Step |
Hyp |
Ref |
Expression |
1 |
|
satf0op.s |
⊢ 𝑆 = ( ∅ Sat ∅ ) |
2 |
|
fveq2 |
⊢ ( 𝑦 = ∅ → ( 𝑆 ‘ 𝑦 ) = ( 𝑆 ‘ ∅ ) ) |
3 |
2
|
eleq2d |
⊢ ( 𝑦 = ∅ → ( 𝑋 ∈ ( 𝑆 ‘ 𝑦 ) ↔ 𝑋 ∈ ( 𝑆 ‘ ∅ ) ) ) |
4 |
2
|
eleq2d |
⊢ ( 𝑦 = ∅ → ( 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑦 ) ↔ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ ∅ ) ) ) |
5 |
4
|
anbi2d |
⊢ ( 𝑦 = ∅ → ( ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑦 ) ) ↔ ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ ∅ ) ) ) ) |
6 |
5
|
exbidv |
⊢ ( 𝑦 = ∅ → ( ∃ 𝑥 ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑦 ) ) ↔ ∃ 𝑥 ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ ∅ ) ) ) ) |
7 |
3 6
|
bibi12d |
⊢ ( 𝑦 = ∅ → ( ( 𝑋 ∈ ( 𝑆 ‘ 𝑦 ) ↔ ∃ 𝑥 ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑦 ) ) ) ↔ ( 𝑋 ∈ ( 𝑆 ‘ ∅ ) ↔ ∃ 𝑥 ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ ∅ ) ) ) ) ) |
8 |
|
fveq2 |
⊢ ( 𝑦 = 𝑧 → ( 𝑆 ‘ 𝑦 ) = ( 𝑆 ‘ 𝑧 ) ) |
9 |
8
|
eleq2d |
⊢ ( 𝑦 = 𝑧 → ( 𝑋 ∈ ( 𝑆 ‘ 𝑦 ) ↔ 𝑋 ∈ ( 𝑆 ‘ 𝑧 ) ) ) |
10 |
8
|
eleq2d |
⊢ ( 𝑦 = 𝑧 → ( 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑦 ) ↔ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑧 ) ) ) |
11 |
10
|
anbi2d |
⊢ ( 𝑦 = 𝑧 → ( ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑦 ) ) ↔ ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑧 ) ) ) ) |
12 |
11
|
exbidv |
⊢ ( 𝑦 = 𝑧 → ( ∃ 𝑥 ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑦 ) ) ↔ ∃ 𝑥 ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑧 ) ) ) ) |
13 |
9 12
|
bibi12d |
⊢ ( 𝑦 = 𝑧 → ( ( 𝑋 ∈ ( 𝑆 ‘ 𝑦 ) ↔ ∃ 𝑥 ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑦 ) ) ) ↔ ( 𝑋 ∈ ( 𝑆 ‘ 𝑧 ) ↔ ∃ 𝑥 ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑧 ) ) ) ) ) |
14 |
|
fveq2 |
⊢ ( 𝑦 = suc 𝑧 → ( 𝑆 ‘ 𝑦 ) = ( 𝑆 ‘ suc 𝑧 ) ) |
15 |
14
|
eleq2d |
⊢ ( 𝑦 = suc 𝑧 → ( 𝑋 ∈ ( 𝑆 ‘ 𝑦 ) ↔ 𝑋 ∈ ( 𝑆 ‘ suc 𝑧 ) ) ) |
16 |
14
|
eleq2d |
⊢ ( 𝑦 = suc 𝑧 → ( 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑦 ) ↔ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ suc 𝑧 ) ) ) |
17 |
16
|
anbi2d |
⊢ ( 𝑦 = suc 𝑧 → ( ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑦 ) ) ↔ ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ suc 𝑧 ) ) ) ) |
18 |
17
|
exbidv |
⊢ ( 𝑦 = suc 𝑧 → ( ∃ 𝑥 ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑦 ) ) ↔ ∃ 𝑥 ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ suc 𝑧 ) ) ) ) |
19 |
15 18
|
bibi12d |
⊢ ( 𝑦 = suc 𝑧 → ( ( 𝑋 ∈ ( 𝑆 ‘ 𝑦 ) ↔ ∃ 𝑥 ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑦 ) ) ) ↔ ( 𝑋 ∈ ( 𝑆 ‘ suc 𝑧 ) ↔ ∃ 𝑥 ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ suc 𝑧 ) ) ) ) ) |
20 |
|
fveq2 |
⊢ ( 𝑦 = 𝑁 → ( 𝑆 ‘ 𝑦 ) = ( 𝑆 ‘ 𝑁 ) ) |
21 |
20
|
eleq2d |
⊢ ( 𝑦 = 𝑁 → ( 𝑋 ∈ ( 𝑆 ‘ 𝑦 ) ↔ 𝑋 ∈ ( 𝑆 ‘ 𝑁 ) ) ) |
22 |
20
|
eleq2d |
⊢ ( 𝑦 = 𝑁 → ( 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑦 ) ↔ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑁 ) ) ) |
23 |
22
|
anbi2d |
⊢ ( 𝑦 = 𝑁 → ( ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑦 ) ) ↔ ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑁 ) ) ) ) |
24 |
23
|
exbidv |
⊢ ( 𝑦 = 𝑁 → ( ∃ 𝑥 ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑦 ) ) ↔ ∃ 𝑥 ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑁 ) ) ) ) |
25 |
21 24
|
bibi12d |
⊢ ( 𝑦 = 𝑁 → ( ( 𝑋 ∈ ( 𝑆 ‘ 𝑦 ) ↔ ∃ 𝑥 ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑦 ) ) ) ↔ ( 𝑋 ∈ ( 𝑆 ‘ 𝑁 ) ↔ ∃ 𝑥 ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑁 ) ) ) ) ) |
26 |
1
|
fveq1i |
⊢ ( 𝑆 ‘ ∅ ) = ( ( ∅ Sat ∅ ) ‘ ∅ ) |
27 |
|
satf00 |
⊢ ( ( ∅ Sat ∅ ) ‘ ∅ ) = { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑦 = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ) } |
28 |
26 27
|
eqtri |
⊢ ( 𝑆 ‘ ∅ ) = { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑦 = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ) } |
29 |
28
|
eleq2i |
⊢ ( 𝑋 ∈ ( 𝑆 ‘ ∅ ) ↔ 𝑋 ∈ { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑦 = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ) } ) |
30 |
|
elopab |
⊢ ( 𝑋 ∈ { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑦 = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ) } ↔ ∃ 𝑥 ∃ 𝑦 ( 𝑋 = 〈 𝑥 , 𝑦 〉 ∧ ( 𝑦 = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ) ) ) |
31 |
|
opeq2 |
⊢ ( 𝑦 = ∅ → 〈 𝑥 , 𝑦 〉 = 〈 𝑥 , ∅ 〉 ) |
32 |
31
|
adantr |
⊢ ( ( 𝑦 = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ) → 〈 𝑥 , 𝑦 〉 = 〈 𝑥 , ∅ 〉 ) |
33 |
32
|
eqeq2d |
⊢ ( ( 𝑦 = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ) → ( 𝑋 = 〈 𝑥 , 𝑦 〉 ↔ 𝑋 = 〈 𝑥 , ∅ 〉 ) ) |
34 |
33
|
biimpd |
⊢ ( ( 𝑦 = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ) → ( 𝑋 = 〈 𝑥 , 𝑦 〉 → 𝑋 = 〈 𝑥 , ∅ 〉 ) ) |
35 |
34
|
impcom |
⊢ ( ( 𝑋 = 〈 𝑥 , 𝑦 〉 ∧ ( 𝑦 = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ) ) → 𝑋 = 〈 𝑥 , ∅ 〉 ) |
36 |
|
eqidd |
⊢ ( 𝑦 = ∅ → ∅ = ∅ ) |
37 |
36
|
anim1i |
⊢ ( ( 𝑦 = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ) → ( ∅ = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ) ) |
38 |
37
|
adantl |
⊢ ( ( 𝑋 = 〈 𝑥 , 𝑦 〉 ∧ ( 𝑦 = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ) ) → ( ∅ = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ) ) |
39 |
|
satf00 |
⊢ ( ( ∅ Sat ∅ ) ‘ ∅ ) = { 〈 𝑦 , 𝑧 〉 ∣ ( 𝑧 = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑦 = ( 𝑖 ∈𝑔 𝑗 ) ) } |
40 |
26 39
|
eqtri |
⊢ ( 𝑆 ‘ ∅ ) = { 〈 𝑦 , 𝑧 〉 ∣ ( 𝑧 = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑦 = ( 𝑖 ∈𝑔 𝑗 ) ) } |
41 |
40
|
eleq2i |
⊢ ( 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ ∅ ) ↔ 〈 𝑥 , ∅ 〉 ∈ { 〈 𝑦 , 𝑧 〉 ∣ ( 𝑧 = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑦 = ( 𝑖 ∈𝑔 𝑗 ) ) } ) |
42 |
|
vex |
⊢ 𝑥 ∈ V |
43 |
|
0ex |
⊢ ∅ ∈ V |
44 |
|
eqeq1 |
⊢ ( 𝑧 = ∅ → ( 𝑧 = ∅ ↔ ∅ = ∅ ) ) |
45 |
|
eqeq1 |
⊢ ( 𝑦 = 𝑥 → ( 𝑦 = ( 𝑖 ∈𝑔 𝑗 ) ↔ 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ) ) |
46 |
45
|
2rexbidv |
⊢ ( 𝑦 = 𝑥 → ( ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑦 = ( 𝑖 ∈𝑔 𝑗 ) ↔ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ) ) |
47 |
44 46
|
bi2anan9r |
⊢ ( ( 𝑦 = 𝑥 ∧ 𝑧 = ∅ ) → ( ( 𝑧 = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑦 = ( 𝑖 ∈𝑔 𝑗 ) ) ↔ ( ∅ = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ) ) ) |
48 |
42 43 47
|
opelopaba |
⊢ ( 〈 𝑥 , ∅ 〉 ∈ { 〈 𝑦 , 𝑧 〉 ∣ ( 𝑧 = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑦 = ( 𝑖 ∈𝑔 𝑗 ) ) } ↔ ( ∅ = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ) ) |
49 |
41 48
|
bitri |
⊢ ( 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ ∅ ) ↔ ( ∅ = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ) ) |
50 |
38 49
|
sylibr |
⊢ ( ( 𝑋 = 〈 𝑥 , 𝑦 〉 ∧ ( 𝑦 = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ) ) → 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ ∅ ) ) |
51 |
35 50
|
jca |
⊢ ( ( 𝑋 = 〈 𝑥 , 𝑦 〉 ∧ ( 𝑦 = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ) ) → ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ ∅ ) ) ) |
52 |
51
|
exlimiv |
⊢ ( ∃ 𝑦 ( 𝑋 = 〈 𝑥 , 𝑦 〉 ∧ ( 𝑦 = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ) ) → ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ ∅ ) ) ) |
53 |
31
|
eqeq2d |
⊢ ( 𝑦 = ∅ → ( 𝑋 = 〈 𝑥 , 𝑦 〉 ↔ 𝑋 = 〈 𝑥 , ∅ 〉 ) ) |
54 |
|
eqeq1 |
⊢ ( 𝑦 = ∅ → ( 𝑦 = ∅ ↔ ∅ = ∅ ) ) |
55 |
54
|
anbi1d |
⊢ ( 𝑦 = ∅ → ( ( 𝑦 = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ) ↔ ( ∅ = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ) ) ) |
56 |
53 55
|
anbi12d |
⊢ ( 𝑦 = ∅ → ( ( 𝑋 = 〈 𝑥 , 𝑦 〉 ∧ ( 𝑦 = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ) ) ↔ ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ ( ∅ = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ) ) ) ) |
57 |
43 56
|
spcev |
⊢ ( ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ ( ∅ = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ) ) → ∃ 𝑦 ( 𝑋 = 〈 𝑥 , 𝑦 〉 ∧ ( 𝑦 = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ) ) ) |
58 |
49 57
|
sylan2b |
⊢ ( ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ ∅ ) ) → ∃ 𝑦 ( 𝑋 = 〈 𝑥 , 𝑦 〉 ∧ ( 𝑦 = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ) ) ) |
59 |
52 58
|
impbii |
⊢ ( ∃ 𝑦 ( 𝑋 = 〈 𝑥 , 𝑦 〉 ∧ ( 𝑦 = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ) ) ↔ ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ ∅ ) ) ) |
60 |
59
|
exbii |
⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝑋 = 〈 𝑥 , 𝑦 〉 ∧ ( 𝑦 = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ) ) ↔ ∃ 𝑥 ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ ∅ ) ) ) |
61 |
29 30 60
|
3bitri |
⊢ ( 𝑋 ∈ ( 𝑆 ‘ ∅ ) ↔ ∃ 𝑥 ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ ∅ ) ) ) |
62 |
1
|
satf0suc |
⊢ ( 𝑧 ∈ ω → ( 𝑆 ‘ suc 𝑧 ) = ( ( 𝑆 ‘ 𝑧 ) ∪ { 〈 𝑎 , 𝑏 〉 ∣ ( 𝑏 = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) } ) ) |
63 |
62
|
eleq2d |
⊢ ( 𝑧 ∈ ω → ( 𝑋 ∈ ( 𝑆 ‘ suc 𝑧 ) ↔ 𝑋 ∈ ( ( 𝑆 ‘ 𝑧 ) ∪ { 〈 𝑎 , 𝑏 〉 ∣ ( 𝑏 = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) } ) ) ) |
64 |
|
elun |
⊢ ( 𝑋 ∈ ( ( 𝑆 ‘ 𝑧 ) ∪ { 〈 𝑎 , 𝑏 〉 ∣ ( 𝑏 = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) } ) ↔ ( 𝑋 ∈ ( 𝑆 ‘ 𝑧 ) ∨ 𝑋 ∈ { 〈 𝑎 , 𝑏 〉 ∣ ( 𝑏 = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) } ) ) |
65 |
64
|
a1i |
⊢ ( 𝑧 ∈ ω → ( 𝑋 ∈ ( ( 𝑆 ‘ 𝑧 ) ∪ { 〈 𝑎 , 𝑏 〉 ∣ ( 𝑏 = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) } ) ↔ ( 𝑋 ∈ ( 𝑆 ‘ 𝑧 ) ∨ 𝑋 ∈ { 〈 𝑎 , 𝑏 〉 ∣ ( 𝑏 = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) } ) ) ) |
66 |
|
elopab |
⊢ ( 𝑋 ∈ { 〈 𝑎 , 𝑏 〉 ∣ ( 𝑏 = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) } ↔ ∃ 𝑎 ∃ 𝑏 ( 𝑋 = 〈 𝑎 , 𝑏 〉 ∧ ( 𝑏 = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ) |
67 |
66
|
a1i |
⊢ ( 𝑧 ∈ ω → ( 𝑋 ∈ { 〈 𝑎 , 𝑏 〉 ∣ ( 𝑏 = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) } ↔ ∃ 𝑎 ∃ 𝑏 ( 𝑋 = 〈 𝑎 , 𝑏 〉 ∧ ( 𝑏 = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ) ) |
68 |
67
|
orbi2d |
⊢ ( 𝑧 ∈ ω → ( ( 𝑋 ∈ ( 𝑆 ‘ 𝑧 ) ∨ 𝑋 ∈ { 〈 𝑎 , 𝑏 〉 ∣ ( 𝑏 = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) } ) ↔ ( 𝑋 ∈ ( 𝑆 ‘ 𝑧 ) ∨ ∃ 𝑎 ∃ 𝑏 ( 𝑋 = 〈 𝑎 , 𝑏 〉 ∧ ( 𝑏 = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ) ) ) |
69 |
63 65 68
|
3bitrd |
⊢ ( 𝑧 ∈ ω → ( 𝑋 ∈ ( 𝑆 ‘ suc 𝑧 ) ↔ ( 𝑋 ∈ ( 𝑆 ‘ 𝑧 ) ∨ ∃ 𝑎 ∃ 𝑏 ( 𝑋 = 〈 𝑎 , 𝑏 〉 ∧ ( 𝑏 = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ) ) ) |
70 |
69
|
adantr |
⊢ ( ( 𝑧 ∈ ω ∧ ( 𝑋 ∈ ( 𝑆 ‘ 𝑧 ) ↔ ∃ 𝑥 ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑧 ) ) ) ) → ( 𝑋 ∈ ( 𝑆 ‘ suc 𝑧 ) ↔ ( 𝑋 ∈ ( 𝑆 ‘ 𝑧 ) ∨ ∃ 𝑎 ∃ 𝑏 ( 𝑋 = 〈 𝑎 , 𝑏 〉 ∧ ( 𝑏 = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ) ) ) |
71 |
|
simpr |
⊢ ( ( 𝑧 ∈ ω ∧ ( 𝑋 ∈ ( 𝑆 ‘ 𝑧 ) ↔ ∃ 𝑥 ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑧 ) ) ) ) → ( 𝑋 ∈ ( 𝑆 ‘ 𝑧 ) ↔ ∃ 𝑥 ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑧 ) ) ) ) |
72 |
|
opeq2 |
⊢ ( 𝑏 = ∅ → 〈 𝑎 , 𝑏 〉 = 〈 𝑎 , ∅ 〉 ) |
73 |
72
|
eqeq2d |
⊢ ( 𝑏 = ∅ → ( 𝑋 = 〈 𝑎 , 𝑏 〉 ↔ 𝑋 = 〈 𝑎 , ∅ 〉 ) ) |
74 |
73
|
biimpd |
⊢ ( 𝑏 = ∅ → ( 𝑋 = 〈 𝑎 , 𝑏 〉 → 𝑋 = 〈 𝑎 , ∅ 〉 ) ) |
75 |
74
|
adantr |
⊢ ( ( 𝑏 = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) → ( 𝑋 = 〈 𝑎 , 𝑏 〉 → 𝑋 = 〈 𝑎 , ∅ 〉 ) ) |
76 |
75
|
impcom |
⊢ ( ( 𝑋 = 〈 𝑎 , 𝑏 〉 ∧ ( 𝑏 = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) → 𝑋 = 〈 𝑎 , ∅ 〉 ) |
77 |
|
eqidd |
⊢ ( ( 𝑋 = 〈 𝑎 , 𝑏 〉 ∧ ( 𝑏 = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) → ∅ = ∅ ) |
78 |
|
simpr |
⊢ ( ( 𝑏 = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) → ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) |
79 |
78
|
adantl |
⊢ ( ( 𝑋 = 〈 𝑎 , 𝑏 〉 ∧ ( 𝑏 = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) → ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) |
80 |
77 79
|
jca |
⊢ ( ( 𝑋 = 〈 𝑎 , 𝑏 〉 ∧ ( 𝑏 = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) → ( ∅ = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) |
81 |
76 80
|
jca |
⊢ ( ( 𝑋 = 〈 𝑎 , 𝑏 〉 ∧ ( 𝑏 = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) → ( 𝑋 = 〈 𝑎 , ∅ 〉 ∧ ( ∅ = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ) |
82 |
81
|
exlimiv |
⊢ ( ∃ 𝑏 ( 𝑋 = 〈 𝑎 , 𝑏 〉 ∧ ( 𝑏 = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) → ( 𝑋 = 〈 𝑎 , ∅ 〉 ∧ ( ∅ = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ) |
83 |
|
eqeq1 |
⊢ ( 𝑏 = ∅ → ( 𝑏 = ∅ ↔ ∅ = ∅ ) ) |
84 |
83
|
anbi1d |
⊢ ( 𝑏 = ∅ → ( ( 𝑏 = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ↔ ( ∅ = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ) |
85 |
73 84
|
anbi12d |
⊢ ( 𝑏 = ∅ → ( ( 𝑋 = 〈 𝑎 , 𝑏 〉 ∧ ( 𝑏 = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ↔ ( 𝑋 = 〈 𝑎 , ∅ 〉 ∧ ( ∅ = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ) ) |
86 |
43 85
|
spcev |
⊢ ( ( 𝑋 = 〈 𝑎 , ∅ 〉 ∧ ( ∅ = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) → ∃ 𝑏 ( 𝑋 = 〈 𝑎 , 𝑏 〉 ∧ ( 𝑏 = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ) |
87 |
82 86
|
impbii |
⊢ ( ∃ 𝑏 ( 𝑋 = 〈 𝑎 , 𝑏 〉 ∧ ( 𝑏 = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ↔ ( 𝑋 = 〈 𝑎 , ∅ 〉 ∧ ( ∅ = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ) |
88 |
87
|
exbii |
⊢ ( ∃ 𝑎 ∃ 𝑏 ( 𝑋 = 〈 𝑎 , 𝑏 〉 ∧ ( 𝑏 = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ↔ ∃ 𝑎 ( 𝑋 = 〈 𝑎 , ∅ 〉 ∧ ( ∅ = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ) |
89 |
88
|
a1i |
⊢ ( 𝑧 ∈ ω → ( ∃ 𝑎 ∃ 𝑏 ( 𝑋 = 〈 𝑎 , 𝑏 〉 ∧ ( 𝑏 = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ↔ ∃ 𝑎 ( 𝑋 = 〈 𝑎 , ∅ 〉 ∧ ( ∅ = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ) ) |
90 |
|
opeq1 |
⊢ ( 𝑥 = 𝑎 → 〈 𝑥 , ∅ 〉 = 〈 𝑎 , ∅ 〉 ) |
91 |
90
|
eqeq2d |
⊢ ( 𝑥 = 𝑎 → ( 𝑋 = 〈 𝑥 , ∅ 〉 ↔ 𝑋 = 〈 𝑎 , ∅ 〉 ) ) |
92 |
|
eqeq1 |
⊢ ( 𝑥 = 𝑎 → ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ↔ 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ) |
93 |
92
|
rexbidv |
⊢ ( 𝑥 = 𝑎 → ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ↔ ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ) |
94 |
|
eqeq1 |
⊢ ( 𝑥 = 𝑎 → ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ↔ 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) |
95 |
94
|
rexbidv |
⊢ ( 𝑥 = 𝑎 → ( ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ↔ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) |
96 |
93 95
|
orbi12d |
⊢ ( 𝑥 = 𝑎 → ( ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ↔ ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) |
97 |
96
|
rexbidv |
⊢ ( 𝑥 = 𝑎 → ( ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ↔ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) |
98 |
97
|
anbi2d |
⊢ ( 𝑥 = 𝑎 → ( ( ∅ = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ↔ ( ∅ = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ) |
99 |
91 98
|
anbi12d |
⊢ ( 𝑥 = 𝑎 → ( ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ ( ∅ = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ↔ ( 𝑋 = 〈 𝑎 , ∅ 〉 ∧ ( ∅ = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ) ) |
100 |
99
|
cbvexvw |
⊢ ( ∃ 𝑥 ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ ( ∅ = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ↔ ∃ 𝑎 ( 𝑋 = 〈 𝑎 , ∅ 〉 ∧ ( ∅ = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ) |
101 |
89 100
|
bitr4di |
⊢ ( 𝑧 ∈ ω → ( ∃ 𝑎 ∃ 𝑏 ( 𝑋 = 〈 𝑎 , 𝑏 〉 ∧ ( 𝑏 = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ↔ ∃ 𝑥 ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ ( ∅ = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ) ) |
102 |
101
|
adantr |
⊢ ( ( 𝑧 ∈ ω ∧ ( 𝑋 ∈ ( 𝑆 ‘ 𝑧 ) ↔ ∃ 𝑥 ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑧 ) ) ) ) → ( ∃ 𝑎 ∃ 𝑏 ( 𝑋 = 〈 𝑎 , 𝑏 〉 ∧ ( 𝑏 = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ↔ ∃ 𝑥 ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ ( ∅ = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ) ) |
103 |
71 102
|
orbi12d |
⊢ ( ( 𝑧 ∈ ω ∧ ( 𝑋 ∈ ( 𝑆 ‘ 𝑧 ) ↔ ∃ 𝑥 ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑧 ) ) ) ) → ( ( 𝑋 ∈ ( 𝑆 ‘ 𝑧 ) ∨ ∃ 𝑎 ∃ 𝑏 ( 𝑋 = 〈 𝑎 , 𝑏 〉 ∧ ( 𝑏 = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ) ↔ ( ∃ 𝑥 ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑧 ) ) ∨ ∃ 𝑥 ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ ( ∅ = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ) ) ) |
104 |
|
19.43 |
⊢ ( ∃ 𝑥 ( ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑧 ) ) ∨ ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ ( ∅ = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ) ↔ ( ∃ 𝑥 ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑧 ) ) ∨ ∃ 𝑥 ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ ( ∅ = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ) ) |
105 |
|
andi |
⊢ ( ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ ( 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑧 ) ∨ ( ∅ = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ) ↔ ( ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑧 ) ) ∨ ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ ( ∅ = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ) ) |
106 |
105
|
bicomi |
⊢ ( ( ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑧 ) ) ∨ ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ ( ∅ = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ) ↔ ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ ( 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑧 ) ∨ ( ∅ = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ) ) |
107 |
106
|
exbii |
⊢ ( ∃ 𝑥 ( ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑧 ) ) ∨ ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ ( ∅ = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ) ↔ ∃ 𝑥 ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ ( 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑧 ) ∨ ( ∅ = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ) ) |
108 |
104 107
|
bitr3i |
⊢ ( ( ∃ 𝑥 ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑧 ) ) ∨ ∃ 𝑥 ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ ( ∅ = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ) ↔ ∃ 𝑥 ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ ( 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑧 ) ∨ ( ∅ = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ) ) |
109 |
103 108
|
bitrdi |
⊢ ( ( 𝑧 ∈ ω ∧ ( 𝑋 ∈ ( 𝑆 ‘ 𝑧 ) ↔ ∃ 𝑥 ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑧 ) ) ) ) → ( ( 𝑋 ∈ ( 𝑆 ‘ 𝑧 ) ∨ ∃ 𝑎 ∃ 𝑏 ( 𝑋 = 〈 𝑎 , 𝑏 〉 ∧ ( 𝑏 = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ) ↔ ∃ 𝑥 ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ ( 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑧 ) ∨ ( ∅ = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ) ) ) |
110 |
62
|
eleq2d |
⊢ ( 𝑧 ∈ ω → ( 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ suc 𝑧 ) ↔ 〈 𝑥 , ∅ 〉 ∈ ( ( 𝑆 ‘ 𝑧 ) ∪ { 〈 𝑎 , 𝑏 〉 ∣ ( 𝑏 = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) } ) ) ) |
111 |
|
elun |
⊢ ( 〈 𝑥 , ∅ 〉 ∈ ( ( 𝑆 ‘ 𝑧 ) ∪ { 〈 𝑎 , 𝑏 〉 ∣ ( 𝑏 = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) } ) ↔ ( 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑧 ) ∨ 〈 𝑥 , ∅ 〉 ∈ { 〈 𝑎 , 𝑏 〉 ∣ ( 𝑏 = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) } ) ) |
112 |
|
eqeq1 |
⊢ ( 𝑎 = 𝑥 → ( 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ↔ 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ) |
113 |
112
|
rexbidv |
⊢ ( 𝑎 = 𝑥 → ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ↔ ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ) |
114 |
|
eqeq1 |
⊢ ( 𝑎 = 𝑥 → ( 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ↔ 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) |
115 |
114
|
rexbidv |
⊢ ( 𝑎 = 𝑥 → ( ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ↔ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) |
116 |
113 115
|
orbi12d |
⊢ ( 𝑎 = 𝑥 → ( ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ↔ ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) |
117 |
116
|
rexbidv |
⊢ ( 𝑎 = 𝑥 → ( ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ↔ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) |
118 |
83 117
|
bi2anan9r |
⊢ ( ( 𝑎 = 𝑥 ∧ 𝑏 = ∅ ) → ( ( 𝑏 = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ↔ ( ∅ = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ) |
119 |
42 43 118
|
opelopaba |
⊢ ( 〈 𝑥 , ∅ 〉 ∈ { 〈 𝑎 , 𝑏 〉 ∣ ( 𝑏 = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) } ↔ ( ∅ = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) |
120 |
119
|
orbi2i |
⊢ ( ( 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑧 ) ∨ 〈 𝑥 , ∅ 〉 ∈ { 〈 𝑎 , 𝑏 〉 ∣ ( 𝑏 = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) } ) ↔ ( 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑧 ) ∨ ( ∅ = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ) |
121 |
111 120
|
bitri |
⊢ ( 〈 𝑥 , ∅ 〉 ∈ ( ( 𝑆 ‘ 𝑧 ) ∪ { 〈 𝑎 , 𝑏 〉 ∣ ( 𝑏 = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑎 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑎 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) } ) ↔ ( 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑧 ) ∨ ( ∅ = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ) |
122 |
110 121
|
bitrdi |
⊢ ( 𝑧 ∈ ω → ( 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ suc 𝑧 ) ↔ ( 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑧 ) ∨ ( ∅ = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ) ) |
123 |
122
|
anbi2d |
⊢ ( 𝑧 ∈ ω → ( ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ suc 𝑧 ) ) ↔ ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ ( 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑧 ) ∨ ( ∅ = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ) ) ) |
124 |
123
|
exbidv |
⊢ ( 𝑧 ∈ ω → ( ∃ 𝑥 ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ suc 𝑧 ) ) ↔ ∃ 𝑥 ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ ( 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑧 ) ∨ ( ∅ = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ) ) ) |
125 |
124
|
bicomd |
⊢ ( 𝑧 ∈ ω → ( ∃ 𝑥 ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ ( 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑧 ) ∨ ( ∅ = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ) ↔ ∃ 𝑥 ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ suc 𝑧 ) ) ) ) |
126 |
125
|
adantr |
⊢ ( ( 𝑧 ∈ ω ∧ ( 𝑋 ∈ ( 𝑆 ‘ 𝑧 ) ↔ ∃ 𝑥 ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑧 ) ) ) ) → ( ∃ 𝑥 ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ ( 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑧 ) ∨ ( ∅ = ∅ ∧ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑧 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑧 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ) ↔ ∃ 𝑥 ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ suc 𝑧 ) ) ) ) |
127 |
70 109 126
|
3bitrd |
⊢ ( ( 𝑧 ∈ ω ∧ ( 𝑋 ∈ ( 𝑆 ‘ 𝑧 ) ↔ ∃ 𝑥 ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑧 ) ) ) ) → ( 𝑋 ∈ ( 𝑆 ‘ suc 𝑧 ) ↔ ∃ 𝑥 ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ suc 𝑧 ) ) ) ) |
128 |
127
|
ex |
⊢ ( 𝑧 ∈ ω → ( ( 𝑋 ∈ ( 𝑆 ‘ 𝑧 ) ↔ ∃ 𝑥 ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑧 ) ) ) → ( 𝑋 ∈ ( 𝑆 ‘ suc 𝑧 ) ↔ ∃ 𝑥 ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ suc 𝑧 ) ) ) ) ) |
129 |
7 13 19 25 61 128
|
finds |
⊢ ( 𝑁 ∈ ω → ( 𝑋 ∈ ( 𝑆 ‘ 𝑁 ) ↔ ∃ 𝑥 ( 𝑋 = 〈 𝑥 , ∅ 〉 ∧ 〈 𝑥 , ∅ 〉 ∈ ( 𝑆 ‘ 𝑁 ) ) ) ) |