Step |
Hyp |
Ref |
Expression |
1 |
|
satfvsucsuc.s |
⊢ 𝑆 = ( 𝑀 Sat 𝐸 ) |
2 |
|
satfvsucsuc.a |
⊢ 𝐴 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) |
3 |
|
satfvsucsuc.b |
⊢ 𝐵 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } |
4 |
|
peano2 |
⊢ ( 𝑁 ∈ ω → suc 𝑁 ∈ ω ) |
5 |
1
|
satfvsuc |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ suc 𝑁 ∈ ω ) → ( 𝑆 ‘ suc suc 𝑁 ) = ( ( 𝑆 ‘ suc 𝑁 ) ∪ { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) } ) ) |
6 |
4 5
|
syl3an3 |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) → ( 𝑆 ‘ suc suc 𝑁 ) = ( ( 𝑆 ‘ suc 𝑁 ) ∪ { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) } ) ) |
7 |
|
orc |
⊢ ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) → ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ∨ ∃ 𝑥 ∃ 𝑦 ( 𝑠 = 〈 𝑥 , 𝑦 〉 ∧ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ) ) ) |
8 |
7
|
a1i |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) → ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) → ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ∨ ∃ 𝑥 ∃ 𝑦 ( 𝑠 = 〈 𝑥 , 𝑦 〉 ∧ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ) ) ) ) |
9 |
2
|
eqeq2i |
⊢ ( 𝑦 = 𝐴 ↔ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) |
10 |
9
|
anbi2i |
⊢ ( ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ↔ ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ) |
11 |
10
|
rexbii |
⊢ ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ↔ ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ) |
12 |
3
|
eqeq2i |
⊢ ( 𝑦 = 𝐵 ↔ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) |
13 |
12
|
anbi2i |
⊢ ( ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ↔ ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) |
14 |
13
|
rexbii |
⊢ ( ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ↔ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) |
15 |
11 14
|
orbi12i |
⊢ ( ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ↔ ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) ) |
16 |
15
|
rexbii |
⊢ ( ∃ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ↔ ∃ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) ) |
17 |
16
|
bicomi |
⊢ ( ∃ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) ↔ ∃ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ) |
18 |
|
3simpa |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) → ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) |
19 |
4
|
ancri |
⊢ ( 𝑁 ∈ ω → ( suc 𝑁 ∈ ω ∧ 𝑁 ∈ ω ) ) |
20 |
19
|
3ad2ant3 |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) → ( suc 𝑁 ∈ ω ∧ 𝑁 ∈ ω ) ) |
21 |
18 20
|
jca |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) → ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ∧ ( suc 𝑁 ∈ ω ∧ 𝑁 ∈ ω ) ) ) |
22 |
|
sssucid |
⊢ 𝑁 ⊆ suc 𝑁 |
23 |
22
|
a1i |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) ∧ 𝑠 = 〈 𝑥 , 𝑦 〉 ) → 𝑁 ⊆ suc 𝑁 ) |
24 |
1
|
satfsschain |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ∧ ( suc 𝑁 ∈ ω ∧ 𝑁 ∈ ω ) ) → ( 𝑁 ⊆ suc 𝑁 → ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ) |
25 |
24
|
imp |
⊢ ( ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ∧ ( suc 𝑁 ∈ ω ∧ 𝑁 ∈ ω ) ) ∧ 𝑁 ⊆ suc 𝑁 ) → ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) |
26 |
21 23 25
|
syl2an2r |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) ∧ 𝑠 = 〈 𝑥 , 𝑦 〉 ) → ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) |
27 |
|
undif |
⊢ ( ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ↔ ( ( 𝑆 ‘ 𝑁 ) ∪ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) = ( 𝑆 ‘ suc 𝑁 ) ) |
28 |
26 27
|
sylib |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) ∧ 𝑠 = 〈 𝑥 , 𝑦 〉 ) → ( ( 𝑆 ‘ 𝑁 ) ∪ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) = ( 𝑆 ‘ suc 𝑁 ) ) |
29 |
28
|
eqcomd |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) ∧ 𝑠 = 〈 𝑥 , 𝑦 〉 ) → ( 𝑆 ‘ suc 𝑁 ) = ( ( 𝑆 ‘ 𝑁 ) ∪ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) |
30 |
29
|
rexeqdv |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) ∧ 𝑠 = 〈 𝑥 , 𝑦 〉 ) → ( ∃ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ↔ ∃ 𝑢 ∈ ( ( 𝑆 ‘ 𝑁 ) ∪ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ) ) |
31 |
|
rexun |
⊢ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ 𝑁 ) ∪ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ↔ ( ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ) ) |
32 |
30 31
|
bitrdi |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) ∧ 𝑠 = 〈 𝑥 , 𝑦 〉 ) → ( ∃ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ↔ ( ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ) ) ) |
33 |
17 32
|
syl5bb |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) ∧ 𝑠 = 〈 𝑥 , 𝑦 〉 ) → ( ∃ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) ↔ ( ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ) ) ) |
34 |
|
r19.43 |
⊢ ( ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ↔ ( ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ) |
35 |
22
|
a1i |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) → 𝑁 ⊆ suc 𝑁 ) |
36 |
21 35
|
jca |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) → ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ∧ ( suc 𝑁 ∈ ω ∧ 𝑁 ∈ ω ) ) ∧ 𝑁 ⊆ suc 𝑁 ) ) |
37 |
36 25
|
syl |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) → ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) |
38 |
37
|
adantr |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) ∧ 𝑠 = 〈 𝑥 , 𝑦 〉 ) → ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) |
39 |
38 27
|
sylib |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) ∧ 𝑠 = 〈 𝑥 , 𝑦 〉 ) → ( ( 𝑆 ‘ 𝑁 ) ∪ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) = ( 𝑆 ‘ suc 𝑁 ) ) |
40 |
39
|
eqcomd |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) ∧ 𝑠 = 〈 𝑥 , 𝑦 〉 ) → ( 𝑆 ‘ suc 𝑁 ) = ( ( 𝑆 ‘ 𝑁 ) ∪ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) |
41 |
40
|
rexeqdv |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) ∧ 𝑠 = 〈 𝑥 , 𝑦 〉 ) → ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ↔ ∃ 𝑣 ∈ ( ( 𝑆 ‘ 𝑁 ) ∪ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ) |
42 |
|
rexun |
⊢ ( ∃ 𝑣 ∈ ( ( 𝑆 ‘ 𝑁 ) ∪ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ↔ ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ) |
43 |
41 42
|
bitrdi |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) ∧ 𝑠 = 〈 𝑥 , 𝑦 〉 ) → ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ↔ ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ) ) |
44 |
43
|
rexbidv |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) ∧ 𝑠 = 〈 𝑥 , 𝑦 〉 ) → ( ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ↔ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ) ) |
45 |
44
|
orbi1d |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) ∧ 𝑠 = 〈 𝑥 , 𝑦 〉 ) → ( ( ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ↔ ( ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ) ) |
46 |
|
r19.43 |
⊢ ( ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ↔ ( ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ) |
47 |
46
|
orbi1i |
⊢ ( ( ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ↔ ( ( ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ) |
48 |
|
or32 |
⊢ ( ( ( ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ↔ ( ( ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ) |
49 |
|
r19.43 |
⊢ ( ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ↔ ( ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ) |
50 |
49
|
bicomi |
⊢ ( ( ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ↔ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ) |
51 |
50
|
orbi1i |
⊢ ( ( ( ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ↔ ( ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ) |
52 |
48 51
|
bitri |
⊢ ( ( ( ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ↔ ( ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ) |
53 |
47 52
|
bitri |
⊢ ( ( ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ↔ ( ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ) |
54 |
45 53
|
bitrdi |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) ∧ 𝑠 = 〈 𝑥 , 𝑦 〉 ) → ( ( ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ↔ ( ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ) ) |
55 |
34 54
|
syl5bb |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) ∧ 𝑠 = 〈 𝑥 , 𝑦 〉 ) → ( ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ↔ ( ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ) ) |
56 |
|
animorr |
⊢ ( ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) ∧ 𝑠 = 〈 𝑥 , 𝑦 〉 ) ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ) → ( 〈 𝑥 , 𝑦 〉 ∈ ( 𝑆 ‘ 𝑁 ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ) ) |
57 |
1
|
satfvsuc |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) → ( 𝑆 ‘ suc 𝑁 ) = ( ( 𝑆 ‘ 𝑁 ) ∪ { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) } ) ) |
58 |
57
|
eleq2d |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) → ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ↔ 𝑠 ∈ ( ( 𝑆 ‘ 𝑁 ) ∪ { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) } ) ) ) |
59 |
58
|
adantr |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) ∧ 𝑠 = 〈 𝑥 , 𝑦 〉 ) → ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ↔ 𝑠 ∈ ( ( 𝑆 ‘ 𝑁 ) ∪ { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) } ) ) ) |
60 |
|
eleq1 |
⊢ ( 𝑠 = 〈 𝑥 , 𝑦 〉 → ( 𝑠 ∈ ( ( 𝑆 ‘ 𝑁 ) ∪ { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) } ) ↔ 〈 𝑥 , 𝑦 〉 ∈ ( ( 𝑆 ‘ 𝑁 ) ∪ { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) } ) ) ) |
61 |
60
|
adantl |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) ∧ 𝑠 = 〈 𝑥 , 𝑦 〉 ) → ( 𝑠 ∈ ( ( 𝑆 ‘ 𝑁 ) ∪ { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) } ) ↔ 〈 𝑥 , 𝑦 〉 ∈ ( ( 𝑆 ‘ 𝑁 ) ∪ { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) } ) ) ) |
62 |
|
elun |
⊢ ( 〈 𝑥 , 𝑦 〉 ∈ ( ( 𝑆 ‘ 𝑁 ) ∪ { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) } ) ↔ ( 〈 𝑥 , 𝑦 〉 ∈ ( 𝑆 ‘ 𝑁 ) ∨ 〈 𝑥 , 𝑦 〉 ∈ { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) } ) ) |
63 |
|
opabidw |
⊢ ( 〈 𝑥 , 𝑦 〉 ∈ { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) } ↔ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) ) |
64 |
63
|
orbi2i |
⊢ ( ( 〈 𝑥 , 𝑦 〉 ∈ ( 𝑆 ‘ 𝑁 ) ∨ 〈 𝑥 , 𝑦 〉 ∈ { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) } ) ↔ ( 〈 𝑥 , 𝑦 〉 ∈ ( 𝑆 ‘ 𝑁 ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) ) ) |
65 |
62 64
|
bitri |
⊢ ( 〈 𝑥 , 𝑦 〉 ∈ ( ( 𝑆 ‘ 𝑁 ) ∪ { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) } ) ↔ ( 〈 𝑥 , 𝑦 〉 ∈ ( 𝑆 ‘ 𝑁 ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) ) ) |
66 |
61 65
|
bitrdi |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) ∧ 𝑠 = 〈 𝑥 , 𝑦 〉 ) → ( 𝑠 ∈ ( ( 𝑆 ‘ 𝑁 ) ∪ { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) } ) ↔ ( 〈 𝑥 , 𝑦 〉 ∈ ( 𝑆 ‘ 𝑁 ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) ) ) ) |
67 |
59 66
|
bitrd |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) ∧ 𝑠 = 〈 𝑥 , 𝑦 〉 ) → ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ↔ ( 〈 𝑥 , 𝑦 〉 ∈ ( 𝑆 ‘ 𝑁 ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) ) ) ) |
68 |
2
|
eqcomi |
⊢ ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) = 𝐴 |
69 |
68
|
eqeq2i |
⊢ ( 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ↔ 𝑦 = 𝐴 ) |
70 |
69
|
anbi2i |
⊢ ( ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ↔ ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) |
71 |
70
|
rexbii |
⊢ ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ↔ ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) |
72 |
3
|
eqcomi |
⊢ { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } = 𝐵 |
73 |
72
|
eqeq2i |
⊢ ( 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ↔ 𝑦 = 𝐵 ) |
74 |
73
|
anbi2i |
⊢ ( ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ↔ ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) |
75 |
74
|
rexbii |
⊢ ( ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ↔ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) |
76 |
71 75
|
orbi12i |
⊢ ( ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) ↔ ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ) |
77 |
76
|
rexbii |
⊢ ( ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) ↔ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ) |
78 |
77
|
a1i |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) ∧ 𝑠 = 〈 𝑥 , 𝑦 〉 ) → ( ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) ↔ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ) ) |
79 |
78
|
orbi2d |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) ∧ 𝑠 = 〈 𝑥 , 𝑦 〉 ) → ( ( 〈 𝑥 , 𝑦 〉 ∈ ( 𝑆 ‘ 𝑁 ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) ) ↔ ( 〈 𝑥 , 𝑦 〉 ∈ ( 𝑆 ‘ 𝑁 ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ) ) ) |
80 |
67 79
|
bitrd |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) ∧ 𝑠 = 〈 𝑥 , 𝑦 〉 ) → ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ↔ ( 〈 𝑥 , 𝑦 〉 ∈ ( 𝑆 ‘ 𝑁 ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ) ) ) |
81 |
80
|
adantr |
⊢ ( ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) ∧ 𝑠 = 〈 𝑥 , 𝑦 〉 ) ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ) → ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ↔ ( 〈 𝑥 , 𝑦 〉 ∈ ( 𝑆 ‘ 𝑁 ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ) ) ) |
82 |
56 81
|
mpbird |
⊢ ( ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) ∧ 𝑠 = 〈 𝑥 , 𝑦 〉 ) ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ) → 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ) |
83 |
82
|
orcd |
⊢ ( ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) ∧ 𝑠 = 〈 𝑥 , 𝑦 〉 ) ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ) → ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ∨ ( 𝑠 = 〈 𝑥 , 𝑦 〉 ∧ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ) ) ) |
84 |
83
|
ex |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) ∧ 𝑠 = 〈 𝑥 , 𝑦 〉 ) → ( ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) → ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ∨ ( 𝑠 = 〈 𝑥 , 𝑦 〉 ∧ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ) ) ) ) |
85 |
|
simplr |
⊢ ( ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) ∧ 𝑠 = 〈 𝑥 , 𝑦 〉 ) ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) → 𝑠 = 〈 𝑥 , 𝑦 〉 ) |
86 |
|
animorr |
⊢ ( ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) ∧ 𝑠 = 〈 𝑥 , 𝑦 〉 ) ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) → ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ) |
87 |
85 86
|
jca |
⊢ ( ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) ∧ 𝑠 = 〈 𝑥 , 𝑦 〉 ) ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) → ( 𝑠 = 〈 𝑥 , 𝑦 〉 ∧ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ) ) |
88 |
87
|
olcd |
⊢ ( ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) ∧ 𝑠 = 〈 𝑥 , 𝑦 〉 ) ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) → ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ∨ ( 𝑠 = 〈 𝑥 , 𝑦 〉 ∧ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ) ) ) |
89 |
88
|
ex |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) ∧ 𝑠 = 〈 𝑥 , 𝑦 〉 ) → ( ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) → ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ∨ ( 𝑠 = 〈 𝑥 , 𝑦 〉 ∧ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ) ) ) ) |
90 |
84 89
|
jaod |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) ∧ 𝑠 = 〈 𝑥 , 𝑦 〉 ) → ( ( ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) → ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ∨ ( 𝑠 = 〈 𝑥 , 𝑦 〉 ∧ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ) ) ) ) |
91 |
55 90
|
sylbid |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) ∧ 𝑠 = 〈 𝑥 , 𝑦 〉 ) → ( ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) → ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ∨ ( 𝑠 = 〈 𝑥 , 𝑦 〉 ∧ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ) ) ) ) |
92 |
|
simplr |
⊢ ( ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) ∧ 𝑠 = 〈 𝑥 , 𝑦 〉 ) ∧ ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ) → 𝑠 = 〈 𝑥 , 𝑦 〉 ) |
93 |
|
orc |
⊢ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) → ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ) |
94 |
93
|
adantl |
⊢ ( ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) ∧ 𝑠 = 〈 𝑥 , 𝑦 〉 ) ∧ ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ) → ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ) |
95 |
92 94
|
jca |
⊢ ( ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) ∧ 𝑠 = 〈 𝑥 , 𝑦 〉 ) ∧ ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ) → ( 𝑠 = 〈 𝑥 , 𝑦 〉 ∧ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ) ) |
96 |
95
|
olcd |
⊢ ( ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) ∧ 𝑠 = 〈 𝑥 , 𝑦 〉 ) ∧ ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ) → ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ∨ ( 𝑠 = 〈 𝑥 , 𝑦 〉 ∧ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ) ) ) |
97 |
96
|
ex |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) ∧ 𝑠 = 〈 𝑥 , 𝑦 〉 ) → ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) → ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ∨ ( 𝑠 = 〈 𝑥 , 𝑦 〉 ∧ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ) ) ) ) |
98 |
91 97
|
jaod |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) ∧ 𝑠 = 〈 𝑥 , 𝑦 〉 ) → ( ( ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ) → ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ∨ ( 𝑠 = 〈 𝑥 , 𝑦 〉 ∧ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ) ) ) ) |
99 |
33 98
|
sylbid |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) ∧ 𝑠 = 〈 𝑥 , 𝑦 〉 ) → ( ∃ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) → ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ∨ ( 𝑠 = 〈 𝑥 , 𝑦 〉 ∧ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ) ) ) ) |
100 |
99
|
expimpd |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) → ( ( 𝑠 = 〈 𝑥 , 𝑦 〉 ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) ) → ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ∨ ( 𝑠 = 〈 𝑥 , 𝑦 〉 ∧ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ) ) ) ) |
101 |
100
|
2eximdv |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) → ( ∃ 𝑥 ∃ 𝑦 ( 𝑠 = 〈 𝑥 , 𝑦 〉 ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) ) → ∃ 𝑥 ∃ 𝑦 ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ∨ ( 𝑠 = 〈 𝑥 , 𝑦 〉 ∧ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ) ) ) ) |
102 |
|
19.45v |
⊢ ( ∃ 𝑦 ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ∨ ( 𝑠 = 〈 𝑥 , 𝑦 〉 ∧ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ) ) ↔ ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ∨ ∃ 𝑦 ( 𝑠 = 〈 𝑥 , 𝑦 〉 ∧ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ) ) ) |
103 |
102
|
exbii |
⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ∨ ( 𝑠 = 〈 𝑥 , 𝑦 〉 ∧ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ) ) ↔ ∃ 𝑥 ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ∨ ∃ 𝑦 ( 𝑠 = 〈 𝑥 , 𝑦 〉 ∧ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ) ) ) |
104 |
|
19.45v |
⊢ ( ∃ 𝑥 ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ∨ ∃ 𝑦 ( 𝑠 = 〈 𝑥 , 𝑦 〉 ∧ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ) ) ↔ ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ∨ ∃ 𝑥 ∃ 𝑦 ( 𝑠 = 〈 𝑥 , 𝑦 〉 ∧ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ) ) ) |
105 |
103 104
|
bitri |
⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ∨ ( 𝑠 = 〈 𝑥 , 𝑦 〉 ∧ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ) ) ↔ ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ∨ ∃ 𝑥 ∃ 𝑦 ( 𝑠 = 〈 𝑥 , 𝑦 〉 ∧ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ) ) ) |
106 |
101 105
|
syl6ib |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) → ( ∃ 𝑥 ∃ 𝑦 ( 𝑠 = 〈 𝑥 , 𝑦 〉 ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) ) → ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ∨ ∃ 𝑥 ∃ 𝑦 ( 𝑠 = 〈 𝑥 , 𝑦 〉 ∧ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ) ) ) ) |
107 |
8 106
|
jaod |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) → ( ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ∨ ∃ 𝑥 ∃ 𝑦 ( 𝑠 = 〈 𝑥 , 𝑦 〉 ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) ) ) → ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ∨ ∃ 𝑥 ∃ 𝑦 ( 𝑠 = 〈 𝑥 , 𝑦 〉 ∧ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ) ) ) ) |
108 |
|
difss |
⊢ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ⊆ ( 𝑆 ‘ suc 𝑁 ) |
109 |
|
ssrexv |
⊢ ( ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ⊆ ( 𝑆 ‘ suc 𝑁 ) → ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) → ∃ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ) ) |
110 |
108 109
|
ax-mp |
⊢ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) → ∃ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ) |
111 |
110
|
a1i |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) → ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) → ∃ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ) ) |
112 |
111 16
|
syl6ib |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) → ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) → ∃ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) ) ) |
113 |
|
ssrexv |
⊢ ( ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) → ( ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) → ∃ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ) |
114 |
37 113
|
syl |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) → ( ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) → ∃ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ) |
115 |
10
|
2rexbii |
⊢ ( ∃ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ↔ ∃ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ) |
116 |
114 115
|
syl6ib |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) → ( ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) → ∃ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ) ) |
117 |
116
|
imp |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) → ∃ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ) |
118 |
|
ssrexv |
⊢ ( ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ⊆ ( 𝑆 ‘ suc 𝑁 ) → ( ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) → ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ) ) |
119 |
108 118
|
ax-mp |
⊢ ( ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) → ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ) |
120 |
119
|
reximi |
⊢ ( ∃ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) → ∃ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ) |
121 |
117 120
|
syl |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) → ∃ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ) |
122 |
121
|
orcd |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) → ( ∃ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) ) |
123 |
122
|
ex |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) → ( ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) → ( ∃ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) ) ) |
124 |
|
r19.43 |
⊢ ( ∃ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) ↔ ( ∃ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) ) |
125 |
123 124
|
syl6ibr |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) → ( ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) → ∃ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) ) ) |
126 |
112 125
|
jaod |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) → ( ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) → ∃ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) ) ) |
127 |
126
|
anim2d |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) → ( ( 𝑠 = 〈 𝑥 , 𝑦 〉 ∧ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ) → ( 𝑠 = 〈 𝑥 , 𝑦 〉 ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) ) ) ) |
128 |
127
|
2eximdv |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) → ( ∃ 𝑥 ∃ 𝑦 ( 𝑠 = 〈 𝑥 , 𝑦 〉 ∧ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ) → ∃ 𝑥 ∃ 𝑦 ( 𝑠 = 〈 𝑥 , 𝑦 〉 ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) ) ) ) |
129 |
128
|
orim2d |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) → ( ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ∨ ∃ 𝑥 ∃ 𝑦 ( 𝑠 = 〈 𝑥 , 𝑦 〉 ∧ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ) ) → ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ∨ ∃ 𝑥 ∃ 𝑦 ( 𝑠 = 〈 𝑥 , 𝑦 〉 ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) ) ) ) ) |
130 |
107 129
|
impbid |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) → ( ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ∨ ∃ 𝑥 ∃ 𝑦 ( 𝑠 = 〈 𝑥 , 𝑦 〉 ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) ) ) ↔ ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ∨ ∃ 𝑥 ∃ 𝑦 ( 𝑠 = 〈 𝑥 , 𝑦 〉 ∧ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ) ) ) ) |
131 |
|
elun |
⊢ ( 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∪ { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) } ) ↔ ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ∨ 𝑠 ∈ { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) } ) ) |
132 |
|
elopab |
⊢ ( 𝑠 ∈ { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) } ↔ ∃ 𝑥 ∃ 𝑦 ( 𝑠 = 〈 𝑥 , 𝑦 〉 ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) ) ) |
133 |
132
|
orbi2i |
⊢ ( ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ∨ 𝑠 ∈ { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) } ) ↔ ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ∨ ∃ 𝑥 ∃ 𝑦 ( 𝑠 = 〈 𝑥 , 𝑦 〉 ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) ) ) ) |
134 |
131 133
|
bitri |
⊢ ( 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∪ { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) } ) ↔ ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ∨ ∃ 𝑥 ∃ 𝑦 ( 𝑠 = 〈 𝑥 , 𝑦 〉 ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) ) ) ) |
135 |
|
elun |
⊢ ( 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∪ { 〈 𝑥 , 𝑦 〉 ∣ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) } ) ↔ ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ∨ 𝑠 ∈ { 〈 𝑥 , 𝑦 〉 ∣ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) } ) ) |
136 |
|
elopab |
⊢ ( 𝑠 ∈ { 〈 𝑥 , 𝑦 〉 ∣ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) } ↔ ∃ 𝑥 ∃ 𝑦 ( 𝑠 = 〈 𝑥 , 𝑦 〉 ∧ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ) ) |
137 |
136
|
orbi2i |
⊢ ( ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ∨ 𝑠 ∈ { 〈 𝑥 , 𝑦 〉 ∣ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) } ) ↔ ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ∨ ∃ 𝑥 ∃ 𝑦 ( 𝑠 = 〈 𝑥 , 𝑦 〉 ∧ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ) ) ) |
138 |
135 137
|
bitri |
⊢ ( 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∪ { 〈 𝑥 , 𝑦 〉 ∣ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) } ) ↔ ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ∨ ∃ 𝑥 ∃ 𝑦 ( 𝑠 = 〈 𝑥 , 𝑦 〉 ∧ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ) ) ) |
139 |
130 134 138
|
3bitr4g |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) → ( 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∪ { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) } ) ↔ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∪ { 〈 𝑥 , 𝑦 〉 ∣ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) } ) ) ) |
140 |
139
|
eqrdv |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) → ( ( 𝑆 ‘ suc 𝑁 ) ∪ { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) } ) = ( ( 𝑆 ‘ suc 𝑁 ) ∪ { 〈 𝑥 , 𝑦 〉 ∣ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) } ) ) |
141 |
6 140
|
eqtrd |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) → ( 𝑆 ‘ suc suc 𝑁 ) = ( ( 𝑆 ‘ suc 𝑁 ) ∪ { 〈 𝑥 , 𝑦 〉 ∣ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) } ) ) |