Step |
Hyp |
Ref |
Expression |
1 |
|
df-1o |
⊢ 1o = suc ∅ |
2 |
1
|
fveq2i |
⊢ ( Fmla ‘ 1o ) = ( Fmla ‘ suc ∅ ) |
3 |
|
peano1 |
⊢ ∅ ∈ ω |
4 |
|
fmlasuc |
⊢ ( ∅ ∈ ω → ( Fmla ‘ suc ∅ ) = ( ( Fmla ‘ ∅ ) ∪ { 𝑥 ∣ ∃ 𝑝 ∈ ( Fmla ‘ ∅ ) ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝 ⊼𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ) } ) ) |
5 |
3 4
|
ax-mp |
⊢ ( Fmla ‘ suc ∅ ) = ( ( Fmla ‘ ∅ ) ∪ { 𝑥 ∣ ∃ 𝑝 ∈ ( Fmla ‘ ∅ ) ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝 ⊼𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ) } ) |
6 |
|
fmla0xp |
⊢ ( Fmla ‘ ∅ ) = ( { ∅ } × ( ω × ω ) ) |
7 |
|
fmla0 |
⊢ ( Fmla ‘ ∅ ) = { 𝑦 ∈ V ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑦 = ( 𝑖 ∈𝑔 𝑗 ) } |
8 |
7
|
rexeqi |
⊢ ( ∃ 𝑝 ∈ ( Fmla ‘ ∅ ) ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝 ⊼𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ) ↔ ∃ 𝑝 ∈ { 𝑦 ∈ V ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑦 = ( 𝑖 ∈𝑔 𝑗 ) } ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝 ⊼𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ) ) |
9 |
|
eqeq1 |
⊢ ( 𝑦 = 𝑝 → ( 𝑦 = ( 𝑖 ∈𝑔 𝑗 ) ↔ 𝑝 = ( 𝑖 ∈𝑔 𝑗 ) ) ) |
10 |
9
|
2rexbidv |
⊢ ( 𝑦 = 𝑝 → ( ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑦 = ( 𝑖 ∈𝑔 𝑗 ) ↔ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑝 = ( 𝑖 ∈𝑔 𝑗 ) ) ) |
11 |
10
|
elrab |
⊢ ( 𝑝 ∈ { 𝑦 ∈ V ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑦 = ( 𝑖 ∈𝑔 𝑗 ) } ↔ ( 𝑝 ∈ V ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑝 = ( 𝑖 ∈𝑔 𝑗 ) ) ) |
12 |
|
oveq1 |
⊢ ( 𝑝 = ( 𝑖 ∈𝑔 𝑗 ) → ( 𝑝 ⊼𝑔 𝑞 ) = ( ( 𝑖 ∈𝑔 𝑗 ) ⊼𝑔 𝑞 ) ) |
13 |
12
|
eqeq2d |
⊢ ( 𝑝 = ( 𝑖 ∈𝑔 𝑗 ) → ( 𝑥 = ( 𝑝 ⊼𝑔 𝑞 ) ↔ 𝑥 = ( ( 𝑖 ∈𝑔 𝑗 ) ⊼𝑔 𝑞 ) ) ) |
14 |
13
|
rexbidv |
⊢ ( 𝑝 = ( 𝑖 ∈𝑔 𝑗 ) → ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝 ⊼𝑔 𝑞 ) ↔ ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( ( 𝑖 ∈𝑔 𝑗 ) ⊼𝑔 𝑞 ) ) ) |
15 |
|
eqidd |
⊢ ( 𝑝 = ( 𝑖 ∈𝑔 𝑗 ) → 𝑘 = 𝑘 ) |
16 |
|
id |
⊢ ( 𝑝 = ( 𝑖 ∈𝑔 𝑗 ) → 𝑝 = ( 𝑖 ∈𝑔 𝑗 ) ) |
17 |
15 16
|
goaleq12d |
⊢ ( 𝑝 = ( 𝑖 ∈𝑔 𝑗 ) → ∀𝑔 𝑘 𝑝 = ∀𝑔 𝑘 ( 𝑖 ∈𝑔 𝑗 ) ) |
18 |
17
|
eqeq2d |
⊢ ( 𝑝 = ( 𝑖 ∈𝑔 𝑗 ) → ( 𝑥 = ∀𝑔 𝑘 𝑝 ↔ 𝑥 = ∀𝑔 𝑘 ( 𝑖 ∈𝑔 𝑗 ) ) ) |
19 |
18
|
rexbidv |
⊢ ( 𝑝 = ( 𝑖 ∈𝑔 𝑗 ) → ( ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ↔ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 ( 𝑖 ∈𝑔 𝑗 ) ) ) |
20 |
14 19
|
orbi12d |
⊢ ( 𝑝 = ( 𝑖 ∈𝑔 𝑗 ) → ( ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝 ⊼𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ) ↔ ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( ( 𝑖 ∈𝑔 𝑗 ) ⊼𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 ( 𝑖 ∈𝑔 𝑗 ) ) ) ) |
21 |
|
eqeq1 |
⊢ ( 𝑧 = 𝑞 → ( 𝑧 = ( 𝑘 ∈𝑔 𝑙 ) ↔ 𝑞 = ( 𝑘 ∈𝑔 𝑙 ) ) ) |
22 |
21
|
2rexbidv |
⊢ ( 𝑧 = 𝑞 → ( ∃ 𝑘 ∈ ω ∃ 𝑙 ∈ ω 𝑧 = ( 𝑘 ∈𝑔 𝑙 ) ↔ ∃ 𝑘 ∈ ω ∃ 𝑙 ∈ ω 𝑞 = ( 𝑘 ∈𝑔 𝑙 ) ) ) |
23 |
|
fmla0 |
⊢ ( Fmla ‘ ∅ ) = { 𝑧 ∈ V ∣ ∃ 𝑘 ∈ ω ∃ 𝑙 ∈ ω 𝑧 = ( 𝑘 ∈𝑔 𝑙 ) } |
24 |
22 23
|
elrab2 |
⊢ ( 𝑞 ∈ ( Fmla ‘ ∅ ) ↔ ( 𝑞 ∈ V ∧ ∃ 𝑘 ∈ ω ∃ 𝑙 ∈ ω 𝑞 = ( 𝑘 ∈𝑔 𝑙 ) ) ) |
25 |
|
oveq2 |
⊢ ( 𝑞 = ( 𝑘 ∈𝑔 𝑙 ) → ( ( 𝑖 ∈𝑔 𝑗 ) ⊼𝑔 𝑞 ) = ( ( 𝑖 ∈𝑔 𝑗 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) ) |
26 |
25
|
eqeq2d |
⊢ ( 𝑞 = ( 𝑘 ∈𝑔 𝑙 ) → ( 𝑥 = ( ( 𝑖 ∈𝑔 𝑗 ) ⊼𝑔 𝑞 ) ↔ 𝑥 = ( ( 𝑖 ∈𝑔 𝑗 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) ) ) |
27 |
26
|
biimpcd |
⊢ ( 𝑥 = ( ( 𝑖 ∈𝑔 𝑗 ) ⊼𝑔 𝑞 ) → ( 𝑞 = ( 𝑘 ∈𝑔 𝑙 ) → 𝑥 = ( ( 𝑖 ∈𝑔 𝑗 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) ) ) |
28 |
27
|
reximdv |
⊢ ( 𝑥 = ( ( 𝑖 ∈𝑔 𝑗 ) ⊼𝑔 𝑞 ) → ( ∃ 𝑙 ∈ ω 𝑞 = ( 𝑘 ∈𝑔 𝑙 ) → ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖 ∈𝑔 𝑗 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) ) ) |
29 |
28
|
reximdv |
⊢ ( 𝑥 = ( ( 𝑖 ∈𝑔 𝑗 ) ⊼𝑔 𝑞 ) → ( ∃ 𝑘 ∈ ω ∃ 𝑙 ∈ ω 𝑞 = ( 𝑘 ∈𝑔 𝑙 ) → ∃ 𝑘 ∈ ω ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖 ∈𝑔 𝑗 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) ) ) |
30 |
29
|
com12 |
⊢ ( ∃ 𝑘 ∈ ω ∃ 𝑙 ∈ ω 𝑞 = ( 𝑘 ∈𝑔 𝑙 ) → ( 𝑥 = ( ( 𝑖 ∈𝑔 𝑗 ) ⊼𝑔 𝑞 ) → ∃ 𝑘 ∈ ω ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖 ∈𝑔 𝑗 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) ) ) |
31 |
24 30
|
simplbiim |
⊢ ( 𝑞 ∈ ( Fmla ‘ ∅ ) → ( 𝑥 = ( ( 𝑖 ∈𝑔 𝑗 ) ⊼𝑔 𝑞 ) → ∃ 𝑘 ∈ ω ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖 ∈𝑔 𝑗 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) ) ) |
32 |
31
|
rexlimiv |
⊢ ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( ( 𝑖 ∈𝑔 𝑗 ) ⊼𝑔 𝑞 ) → ∃ 𝑘 ∈ ω ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖 ∈𝑔 𝑗 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) ) |
33 |
32
|
orim1i |
⊢ ( ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( ( 𝑖 ∈𝑔 𝑗 ) ⊼𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 ( 𝑖 ∈𝑔 𝑗 ) ) → ( ∃ 𝑘 ∈ ω ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖 ∈𝑔 𝑗 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 ( 𝑖 ∈𝑔 𝑗 ) ) ) |
34 |
|
r19.43 |
⊢ ( ∃ 𝑘 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖 ∈𝑔 𝑗 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑖 ∈𝑔 𝑗 ) ) ↔ ( ∃ 𝑘 ∈ ω ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖 ∈𝑔 𝑗 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 ( 𝑖 ∈𝑔 𝑗 ) ) ) |
35 |
33 34
|
sylibr |
⊢ ( ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( ( 𝑖 ∈𝑔 𝑗 ) ⊼𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 ( 𝑖 ∈𝑔 𝑗 ) ) → ∃ 𝑘 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖 ∈𝑔 𝑗 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑖 ∈𝑔 𝑗 ) ) ) |
36 |
20 35
|
syl6bi |
⊢ ( 𝑝 = ( 𝑖 ∈𝑔 𝑗 ) → ( ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝 ⊼𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ) → ∃ 𝑘 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖 ∈𝑔 𝑗 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑖 ∈𝑔 𝑗 ) ) ) ) |
37 |
36
|
com12 |
⊢ ( ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝 ⊼𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ) → ( 𝑝 = ( 𝑖 ∈𝑔 𝑗 ) → ∃ 𝑘 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖 ∈𝑔 𝑗 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑖 ∈𝑔 𝑗 ) ) ) ) |
38 |
37
|
reximdv |
⊢ ( ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝 ⊼𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ) → ( ∃ 𝑗 ∈ ω 𝑝 = ( 𝑖 ∈𝑔 𝑗 ) → ∃ 𝑗 ∈ ω ∃ 𝑘 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖 ∈𝑔 𝑗 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑖 ∈𝑔 𝑗 ) ) ) ) |
39 |
38
|
reximdv |
⊢ ( ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝 ⊼𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ) → ( ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑝 = ( 𝑖 ∈𝑔 𝑗 ) → ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ∃ 𝑘 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖 ∈𝑔 𝑗 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑖 ∈𝑔 𝑗 ) ) ) ) |
40 |
39
|
com12 |
⊢ ( ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑝 = ( 𝑖 ∈𝑔 𝑗 ) → ( ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝 ⊼𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ) → ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ∃ 𝑘 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖 ∈𝑔 𝑗 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑖 ∈𝑔 𝑗 ) ) ) ) |
41 |
11 40
|
simplbiim |
⊢ ( 𝑝 ∈ { 𝑦 ∈ V ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑦 = ( 𝑖 ∈𝑔 𝑗 ) } → ( ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝 ⊼𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ) → ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ∃ 𝑘 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖 ∈𝑔 𝑗 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑖 ∈𝑔 𝑗 ) ) ) ) |
42 |
41
|
rexlimiv |
⊢ ( ∃ 𝑝 ∈ { 𝑦 ∈ V ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑦 = ( 𝑖 ∈𝑔 𝑗 ) } ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝 ⊼𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ) → ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ∃ 𝑘 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖 ∈𝑔 𝑗 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑖 ∈𝑔 𝑗 ) ) ) |
43 |
|
oveq1 |
⊢ ( 𝑖 = 𝑚 → ( 𝑖 ∈𝑔 𝑗 ) = ( 𝑚 ∈𝑔 𝑗 ) ) |
44 |
43
|
oveq1d |
⊢ ( 𝑖 = 𝑚 → ( ( 𝑖 ∈𝑔 𝑗 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) = ( ( 𝑚 ∈𝑔 𝑗 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) ) |
45 |
44
|
eqeq2d |
⊢ ( 𝑖 = 𝑚 → ( 𝑥 = ( ( 𝑖 ∈𝑔 𝑗 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) ↔ 𝑥 = ( ( 𝑚 ∈𝑔 𝑗 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) ) ) |
46 |
45
|
rexbidv |
⊢ ( 𝑖 = 𝑚 → ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖 ∈𝑔 𝑗 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) ↔ ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑚 ∈𝑔 𝑗 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) ) ) |
47 |
|
eqidd |
⊢ ( 𝑖 = 𝑚 → 𝑘 = 𝑘 ) |
48 |
47 43
|
goaleq12d |
⊢ ( 𝑖 = 𝑚 → ∀𝑔 𝑘 ( 𝑖 ∈𝑔 𝑗 ) = ∀𝑔 𝑘 ( 𝑚 ∈𝑔 𝑗 ) ) |
49 |
48
|
eqeq2d |
⊢ ( 𝑖 = 𝑚 → ( 𝑥 = ∀𝑔 𝑘 ( 𝑖 ∈𝑔 𝑗 ) ↔ 𝑥 = ∀𝑔 𝑘 ( 𝑚 ∈𝑔 𝑗 ) ) ) |
50 |
46 49
|
orbi12d |
⊢ ( 𝑖 = 𝑚 → ( ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖 ∈𝑔 𝑗 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑖 ∈𝑔 𝑗 ) ) ↔ ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑚 ∈𝑔 𝑗 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑚 ∈𝑔 𝑗 ) ) ) ) |
51 |
50
|
rexbidv |
⊢ ( 𝑖 = 𝑚 → ( ∃ 𝑘 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖 ∈𝑔 𝑗 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑖 ∈𝑔 𝑗 ) ) ↔ ∃ 𝑘 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑚 ∈𝑔 𝑗 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑚 ∈𝑔 𝑗 ) ) ) ) |
52 |
|
oveq2 |
⊢ ( 𝑗 = 𝑛 → ( 𝑚 ∈𝑔 𝑗 ) = ( 𝑚 ∈𝑔 𝑛 ) ) |
53 |
52
|
oveq1d |
⊢ ( 𝑗 = 𝑛 → ( ( 𝑚 ∈𝑔 𝑗 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) = ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) ) |
54 |
53
|
eqeq2d |
⊢ ( 𝑗 = 𝑛 → ( 𝑥 = ( ( 𝑚 ∈𝑔 𝑗 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) ↔ 𝑥 = ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) ) ) |
55 |
54
|
rexbidv |
⊢ ( 𝑗 = 𝑛 → ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑚 ∈𝑔 𝑗 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) ↔ ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) ) ) |
56 |
|
eqidd |
⊢ ( 𝑗 = 𝑛 → 𝑘 = 𝑘 ) |
57 |
56 52
|
goaleq12d |
⊢ ( 𝑗 = 𝑛 → ∀𝑔 𝑘 ( 𝑚 ∈𝑔 𝑗 ) = ∀𝑔 𝑘 ( 𝑚 ∈𝑔 𝑛 ) ) |
58 |
57
|
eqeq2d |
⊢ ( 𝑗 = 𝑛 → ( 𝑥 = ∀𝑔 𝑘 ( 𝑚 ∈𝑔 𝑗 ) ↔ 𝑥 = ∀𝑔 𝑘 ( 𝑚 ∈𝑔 𝑛 ) ) ) |
59 |
55 58
|
orbi12d |
⊢ ( 𝑗 = 𝑛 → ( ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑚 ∈𝑔 𝑗 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑚 ∈𝑔 𝑗 ) ) ↔ ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑚 ∈𝑔 𝑛 ) ) ) ) |
60 |
59
|
rexbidv |
⊢ ( 𝑗 = 𝑛 → ( ∃ 𝑘 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑚 ∈𝑔 𝑗 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑚 ∈𝑔 𝑗 ) ) ↔ ∃ 𝑘 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑚 ∈𝑔 𝑛 ) ) ) ) |
61 |
51 60
|
cbvrex2vw |
⊢ ( ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ∃ 𝑘 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖 ∈𝑔 𝑗 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑖 ∈𝑔 𝑗 ) ) ↔ ∃ 𝑚 ∈ ω ∃ 𝑛 ∈ ω ∃ 𝑘 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑚 ∈𝑔 𝑛 ) ) ) |
62 |
|
oveq1 |
⊢ ( 𝑘 = 𝑜 → ( 𝑘 ∈𝑔 𝑙 ) = ( 𝑜 ∈𝑔 𝑙 ) ) |
63 |
62
|
oveq2d |
⊢ ( 𝑘 = 𝑜 → ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) = ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 ( 𝑜 ∈𝑔 𝑙 ) ) ) |
64 |
63
|
eqeq2d |
⊢ ( 𝑘 = 𝑜 → ( 𝑥 = ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) ↔ 𝑥 = ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 ( 𝑜 ∈𝑔 𝑙 ) ) ) ) |
65 |
64
|
rexbidv |
⊢ ( 𝑘 = 𝑜 → ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) ↔ ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 ( 𝑜 ∈𝑔 𝑙 ) ) ) ) |
66 |
|
id |
⊢ ( 𝑘 = 𝑜 → 𝑘 = 𝑜 ) |
67 |
|
eqidd |
⊢ ( 𝑘 = 𝑜 → ( 𝑚 ∈𝑔 𝑛 ) = ( 𝑚 ∈𝑔 𝑛 ) ) |
68 |
66 67
|
goaleq12d |
⊢ ( 𝑘 = 𝑜 → ∀𝑔 𝑘 ( 𝑚 ∈𝑔 𝑛 ) = ∀𝑔 𝑜 ( 𝑚 ∈𝑔 𝑛 ) ) |
69 |
68
|
eqeq2d |
⊢ ( 𝑘 = 𝑜 → ( 𝑥 = ∀𝑔 𝑘 ( 𝑚 ∈𝑔 𝑛 ) ↔ 𝑥 = ∀𝑔 𝑜 ( 𝑚 ∈𝑔 𝑛 ) ) ) |
70 |
65 69
|
orbi12d |
⊢ ( 𝑘 = 𝑜 → ( ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑚 ∈𝑔 𝑛 ) ) ↔ ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 ( 𝑜 ∈𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑜 ( 𝑚 ∈𝑔 𝑛 ) ) ) ) |
71 |
70
|
cbvrexvw |
⊢ ( ∃ 𝑘 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑚 ∈𝑔 𝑛 ) ) ↔ ∃ 𝑜 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 ( 𝑜 ∈𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑜 ( 𝑚 ∈𝑔 𝑛 ) ) ) |
72 |
3
|
ne0ii |
⊢ ω ≠ ∅ |
73 |
|
r19.44zv |
⊢ ( ω ≠ ∅ → ( ∃ 𝑙 ∈ ω ( 𝑥 = ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 ( 𝑜 ∈𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑜 ( 𝑚 ∈𝑔 𝑛 ) ) ↔ ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 ( 𝑜 ∈𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑜 ( 𝑚 ∈𝑔 𝑛 ) ) ) ) |
74 |
72 73
|
ax-mp |
⊢ ( ∃ 𝑙 ∈ ω ( 𝑥 = ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 ( 𝑜 ∈𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑜 ( 𝑚 ∈𝑔 𝑛 ) ) ↔ ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 ( 𝑜 ∈𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑜 ( 𝑚 ∈𝑔 𝑛 ) ) ) |
75 |
|
eqeq1 |
⊢ ( 𝑦 = ( 𝑚 ∈𝑔 𝑛 ) → ( 𝑦 = ( 𝑖 ∈𝑔 𝑗 ) ↔ ( 𝑚 ∈𝑔 𝑛 ) = ( 𝑖 ∈𝑔 𝑗 ) ) ) |
76 |
75
|
2rexbidv |
⊢ ( 𝑦 = ( 𝑚 ∈𝑔 𝑛 ) → ( ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑦 = ( 𝑖 ∈𝑔 𝑗 ) ↔ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑚 ∈𝑔 𝑛 ) = ( 𝑖 ∈𝑔 𝑗 ) ) ) |
77 |
|
ovexd |
⊢ ( ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) ∧ ( 𝑥 = ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 ( 𝑜 ∈𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑜 ( 𝑚 ∈𝑔 𝑛 ) ) ) → ( 𝑚 ∈𝑔 𝑛 ) ∈ V ) |
78 |
|
simpl |
⊢ ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) → 𝑚 ∈ ω ) |
79 |
43
|
eqeq2d |
⊢ ( 𝑖 = 𝑚 → ( ( 𝑚 ∈𝑔 𝑛 ) = ( 𝑖 ∈𝑔 𝑗 ) ↔ ( 𝑚 ∈𝑔 𝑛 ) = ( 𝑚 ∈𝑔 𝑗 ) ) ) |
80 |
79
|
rexbidv |
⊢ ( 𝑖 = 𝑚 → ( ∃ 𝑗 ∈ ω ( 𝑚 ∈𝑔 𝑛 ) = ( 𝑖 ∈𝑔 𝑗 ) ↔ ∃ 𝑗 ∈ ω ( 𝑚 ∈𝑔 𝑛 ) = ( 𝑚 ∈𝑔 𝑗 ) ) ) |
81 |
80
|
adantl |
⊢ ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑖 = 𝑚 ) → ( ∃ 𝑗 ∈ ω ( 𝑚 ∈𝑔 𝑛 ) = ( 𝑖 ∈𝑔 𝑗 ) ↔ ∃ 𝑗 ∈ ω ( 𝑚 ∈𝑔 𝑛 ) = ( 𝑚 ∈𝑔 𝑗 ) ) ) |
82 |
|
simpr |
⊢ ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) → 𝑛 ∈ ω ) |
83 |
52
|
eqeq2d |
⊢ ( 𝑗 = 𝑛 → ( ( 𝑚 ∈𝑔 𝑛 ) = ( 𝑚 ∈𝑔 𝑗 ) ↔ ( 𝑚 ∈𝑔 𝑛 ) = ( 𝑚 ∈𝑔 𝑛 ) ) ) |
84 |
83
|
adantl |
⊢ ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑗 = 𝑛 ) → ( ( 𝑚 ∈𝑔 𝑛 ) = ( 𝑚 ∈𝑔 𝑗 ) ↔ ( 𝑚 ∈𝑔 𝑛 ) = ( 𝑚 ∈𝑔 𝑛 ) ) ) |
85 |
|
eqidd |
⊢ ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) → ( 𝑚 ∈𝑔 𝑛 ) = ( 𝑚 ∈𝑔 𝑛 ) ) |
86 |
82 84 85
|
rspcedvd |
⊢ ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) → ∃ 𝑗 ∈ ω ( 𝑚 ∈𝑔 𝑛 ) = ( 𝑚 ∈𝑔 𝑗 ) ) |
87 |
78 81 86
|
rspcedvd |
⊢ ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) → ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑚 ∈𝑔 𝑛 ) = ( 𝑖 ∈𝑔 𝑗 ) ) |
88 |
87
|
ad5ant12 |
⊢ ( ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) ∧ ( 𝑥 = ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 ( 𝑜 ∈𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑜 ( 𝑚 ∈𝑔 𝑛 ) ) ) → ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑚 ∈𝑔 𝑛 ) = ( 𝑖 ∈𝑔 𝑗 ) ) |
89 |
76 77 88
|
elrabd |
⊢ ( ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) ∧ ( 𝑥 = ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 ( 𝑜 ∈𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑜 ( 𝑚 ∈𝑔 𝑛 ) ) ) → ( 𝑚 ∈𝑔 𝑛 ) ∈ { 𝑦 ∈ V ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑦 = ( 𝑖 ∈𝑔 𝑗 ) } ) |
90 |
|
oveq1 |
⊢ ( 𝑝 = ( 𝑚 ∈𝑔 𝑛 ) → ( 𝑝 ⊼𝑔 𝑞 ) = ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 𝑞 ) ) |
91 |
90
|
eqeq2d |
⊢ ( 𝑝 = ( 𝑚 ∈𝑔 𝑛 ) → ( 𝑥 = ( 𝑝 ⊼𝑔 𝑞 ) ↔ 𝑥 = ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 𝑞 ) ) ) |
92 |
91
|
rexbidv |
⊢ ( 𝑝 = ( 𝑚 ∈𝑔 𝑛 ) → ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝 ⊼𝑔 𝑞 ) ↔ ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 𝑞 ) ) ) |
93 |
|
eqidd |
⊢ ( 𝑝 = ( 𝑚 ∈𝑔 𝑛 ) → 𝑘 = 𝑘 ) |
94 |
|
id |
⊢ ( 𝑝 = ( 𝑚 ∈𝑔 𝑛 ) → 𝑝 = ( 𝑚 ∈𝑔 𝑛 ) ) |
95 |
93 94
|
goaleq12d |
⊢ ( 𝑝 = ( 𝑚 ∈𝑔 𝑛 ) → ∀𝑔 𝑘 𝑝 = ∀𝑔 𝑘 ( 𝑚 ∈𝑔 𝑛 ) ) |
96 |
95
|
eqeq2d |
⊢ ( 𝑝 = ( 𝑚 ∈𝑔 𝑛 ) → ( 𝑥 = ∀𝑔 𝑘 𝑝 ↔ 𝑥 = ∀𝑔 𝑘 ( 𝑚 ∈𝑔 𝑛 ) ) ) |
97 |
96
|
rexbidv |
⊢ ( 𝑝 = ( 𝑚 ∈𝑔 𝑛 ) → ( ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ↔ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 ( 𝑚 ∈𝑔 𝑛 ) ) ) |
98 |
92 97
|
orbi12d |
⊢ ( 𝑝 = ( 𝑚 ∈𝑔 𝑛 ) → ( ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝 ⊼𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ) ↔ ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 ( 𝑚 ∈𝑔 𝑛 ) ) ) ) |
99 |
98
|
adantl |
⊢ ( ( ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) ∧ ( 𝑥 = ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 ( 𝑜 ∈𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑜 ( 𝑚 ∈𝑔 𝑛 ) ) ) ∧ 𝑝 = ( 𝑚 ∈𝑔 𝑛 ) ) → ( ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝 ⊼𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ) ↔ ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 ( 𝑚 ∈𝑔 𝑛 ) ) ) ) |
100 |
|
ovexd |
⊢ ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) → ( 𝑜 ∈𝑔 𝑙 ) ∈ V ) |
101 |
|
simpr |
⊢ ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) → 𝑜 ∈ ω ) |
102 |
101
|
adantr |
⊢ ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) → 𝑜 ∈ ω ) |
103 |
|
oveq1 |
⊢ ( 𝑖 = 𝑜 → ( 𝑖 ∈𝑔 𝑗 ) = ( 𝑜 ∈𝑔 𝑗 ) ) |
104 |
103
|
eqeq2d |
⊢ ( 𝑖 = 𝑜 → ( ( 𝑜 ∈𝑔 𝑙 ) = ( 𝑖 ∈𝑔 𝑗 ) ↔ ( 𝑜 ∈𝑔 𝑙 ) = ( 𝑜 ∈𝑔 𝑗 ) ) ) |
105 |
104
|
rexbidv |
⊢ ( 𝑖 = 𝑜 → ( ∃ 𝑗 ∈ ω ( 𝑜 ∈𝑔 𝑙 ) = ( 𝑖 ∈𝑔 𝑗 ) ↔ ∃ 𝑗 ∈ ω ( 𝑜 ∈𝑔 𝑙 ) = ( 𝑜 ∈𝑔 𝑗 ) ) ) |
106 |
105
|
adantl |
⊢ ( ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) ∧ 𝑖 = 𝑜 ) → ( ∃ 𝑗 ∈ ω ( 𝑜 ∈𝑔 𝑙 ) = ( 𝑖 ∈𝑔 𝑗 ) ↔ ∃ 𝑗 ∈ ω ( 𝑜 ∈𝑔 𝑙 ) = ( 𝑜 ∈𝑔 𝑗 ) ) ) |
107 |
|
simpr |
⊢ ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) → 𝑙 ∈ ω ) |
108 |
|
oveq2 |
⊢ ( 𝑗 = 𝑙 → ( 𝑜 ∈𝑔 𝑗 ) = ( 𝑜 ∈𝑔 𝑙 ) ) |
109 |
108
|
eqeq2d |
⊢ ( 𝑗 = 𝑙 → ( ( 𝑜 ∈𝑔 𝑙 ) = ( 𝑜 ∈𝑔 𝑗 ) ↔ ( 𝑜 ∈𝑔 𝑙 ) = ( 𝑜 ∈𝑔 𝑙 ) ) ) |
110 |
109
|
adantl |
⊢ ( ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) ∧ 𝑗 = 𝑙 ) → ( ( 𝑜 ∈𝑔 𝑙 ) = ( 𝑜 ∈𝑔 𝑗 ) ↔ ( 𝑜 ∈𝑔 𝑙 ) = ( 𝑜 ∈𝑔 𝑙 ) ) ) |
111 |
|
eqidd |
⊢ ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) → ( 𝑜 ∈𝑔 𝑙 ) = ( 𝑜 ∈𝑔 𝑙 ) ) |
112 |
107 110 111
|
rspcedvd |
⊢ ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) → ∃ 𝑗 ∈ ω ( 𝑜 ∈𝑔 𝑙 ) = ( 𝑜 ∈𝑔 𝑗 ) ) |
113 |
102 106 112
|
rspcedvd |
⊢ ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) → ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑜 ∈𝑔 𝑙 ) = ( 𝑖 ∈𝑔 𝑗 ) ) |
114 |
|
eqeq1 |
⊢ ( 𝑝 = ( 𝑜 ∈𝑔 𝑙 ) → ( 𝑝 = ( 𝑖 ∈𝑔 𝑗 ) ↔ ( 𝑜 ∈𝑔 𝑙 ) = ( 𝑖 ∈𝑔 𝑗 ) ) ) |
115 |
114
|
2rexbidv |
⊢ ( 𝑝 = ( 𝑜 ∈𝑔 𝑙 ) → ( ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑝 = ( 𝑖 ∈𝑔 𝑗 ) ↔ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑜 ∈𝑔 𝑙 ) = ( 𝑖 ∈𝑔 𝑗 ) ) ) |
116 |
|
fmla0 |
⊢ ( Fmla ‘ ∅ ) = { 𝑝 ∈ V ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑝 = ( 𝑖 ∈𝑔 𝑗 ) } |
117 |
115 116
|
elrab2 |
⊢ ( ( 𝑜 ∈𝑔 𝑙 ) ∈ ( Fmla ‘ ∅ ) ↔ ( ( 𝑜 ∈𝑔 𝑙 ) ∈ V ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑜 ∈𝑔 𝑙 ) = ( 𝑖 ∈𝑔 𝑗 ) ) ) |
118 |
100 113 117
|
sylanbrc |
⊢ ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) → ( 𝑜 ∈𝑔 𝑙 ) ∈ ( Fmla ‘ ∅ ) ) |
119 |
118
|
adantr |
⊢ ( ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) ∧ 𝑥 = ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 ( 𝑜 ∈𝑔 𝑙 ) ) ) → ( 𝑜 ∈𝑔 𝑙 ) ∈ ( Fmla ‘ ∅ ) ) |
120 |
|
oveq2 |
⊢ ( 𝑞 = ( 𝑜 ∈𝑔 𝑙 ) → ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 𝑞 ) = ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 ( 𝑜 ∈𝑔 𝑙 ) ) ) |
121 |
120
|
eqeq2d |
⊢ ( 𝑞 = ( 𝑜 ∈𝑔 𝑙 ) → ( 𝑥 = ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 𝑞 ) ↔ 𝑥 = ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 ( 𝑜 ∈𝑔 𝑙 ) ) ) ) |
122 |
121
|
adantl |
⊢ ( ( ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) ∧ 𝑥 = ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 ( 𝑜 ∈𝑔 𝑙 ) ) ) ∧ 𝑞 = ( 𝑜 ∈𝑔 𝑙 ) ) → ( 𝑥 = ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 𝑞 ) ↔ 𝑥 = ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 ( 𝑜 ∈𝑔 𝑙 ) ) ) ) |
123 |
|
simpr |
⊢ ( ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) ∧ 𝑥 = ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 ( 𝑜 ∈𝑔 𝑙 ) ) ) → 𝑥 = ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 ( 𝑜 ∈𝑔 𝑙 ) ) ) |
124 |
119 122 123
|
rspcedvd |
⊢ ( ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) ∧ 𝑥 = ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 ( 𝑜 ∈𝑔 𝑙 ) ) ) → ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 𝑞 ) ) |
125 |
124
|
ex |
⊢ ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) → ( 𝑥 = ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 ( 𝑜 ∈𝑔 𝑙 ) ) → ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 𝑞 ) ) ) |
126 |
102
|
adantr |
⊢ ( ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) ∧ 𝑥 = ∀𝑔 𝑜 ( 𝑚 ∈𝑔 𝑛 ) ) → 𝑜 ∈ ω ) |
127 |
69
|
adantl |
⊢ ( ( ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) ∧ 𝑥 = ∀𝑔 𝑜 ( 𝑚 ∈𝑔 𝑛 ) ) ∧ 𝑘 = 𝑜 ) → ( 𝑥 = ∀𝑔 𝑘 ( 𝑚 ∈𝑔 𝑛 ) ↔ 𝑥 = ∀𝑔 𝑜 ( 𝑚 ∈𝑔 𝑛 ) ) ) |
128 |
|
simpr |
⊢ ( ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) ∧ 𝑥 = ∀𝑔 𝑜 ( 𝑚 ∈𝑔 𝑛 ) ) → 𝑥 = ∀𝑔 𝑜 ( 𝑚 ∈𝑔 𝑛 ) ) |
129 |
126 127 128
|
rspcedvd |
⊢ ( ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) ∧ 𝑥 = ∀𝑔 𝑜 ( 𝑚 ∈𝑔 𝑛 ) ) → ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 ( 𝑚 ∈𝑔 𝑛 ) ) |
130 |
129
|
ex |
⊢ ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) → ( 𝑥 = ∀𝑔 𝑜 ( 𝑚 ∈𝑔 𝑛 ) → ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 ( 𝑚 ∈𝑔 𝑛 ) ) ) |
131 |
125 130
|
orim12d |
⊢ ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) → ( ( 𝑥 = ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 ( 𝑜 ∈𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑜 ( 𝑚 ∈𝑔 𝑛 ) ) → ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 ( 𝑚 ∈𝑔 𝑛 ) ) ) ) |
132 |
131
|
imp |
⊢ ( ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) ∧ ( 𝑥 = ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 ( 𝑜 ∈𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑜 ( 𝑚 ∈𝑔 𝑛 ) ) ) → ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 ( 𝑚 ∈𝑔 𝑛 ) ) ) |
133 |
89 99 132
|
rspcedvd |
⊢ ( ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) ∧ ( 𝑥 = ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 ( 𝑜 ∈𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑜 ( 𝑚 ∈𝑔 𝑛 ) ) ) → ∃ 𝑝 ∈ { 𝑦 ∈ V ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑦 = ( 𝑖 ∈𝑔 𝑗 ) } ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝 ⊼𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ) ) |
134 |
133
|
ex |
⊢ ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) → ( ( 𝑥 = ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 ( 𝑜 ∈𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑜 ( 𝑚 ∈𝑔 𝑛 ) ) → ∃ 𝑝 ∈ { 𝑦 ∈ V ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑦 = ( 𝑖 ∈𝑔 𝑗 ) } ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝 ⊼𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ) ) ) |
135 |
134
|
rexlimdva |
⊢ ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) → ( ∃ 𝑙 ∈ ω ( 𝑥 = ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 ( 𝑜 ∈𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑜 ( 𝑚 ∈𝑔 𝑛 ) ) → ∃ 𝑝 ∈ { 𝑦 ∈ V ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑦 = ( 𝑖 ∈𝑔 𝑗 ) } ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝 ⊼𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ) ) ) |
136 |
74 135
|
syl5bir |
⊢ ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) → ( ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 ( 𝑜 ∈𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑜 ( 𝑚 ∈𝑔 𝑛 ) ) → ∃ 𝑝 ∈ { 𝑦 ∈ V ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑦 = ( 𝑖 ∈𝑔 𝑗 ) } ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝 ⊼𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ) ) ) |
137 |
136
|
rexlimdva |
⊢ ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) → ( ∃ 𝑜 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 ( 𝑜 ∈𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑜 ( 𝑚 ∈𝑔 𝑛 ) ) → ∃ 𝑝 ∈ { 𝑦 ∈ V ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑦 = ( 𝑖 ∈𝑔 𝑗 ) } ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝 ⊼𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ) ) ) |
138 |
71 137
|
syl5bi |
⊢ ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) → ( ∃ 𝑘 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑚 ∈𝑔 𝑛 ) ) → ∃ 𝑝 ∈ { 𝑦 ∈ V ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑦 = ( 𝑖 ∈𝑔 𝑗 ) } ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝 ⊼𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ) ) ) |
139 |
138
|
rexlimivv |
⊢ ( ∃ 𝑚 ∈ ω ∃ 𝑛 ∈ ω ∃ 𝑘 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑚 ∈𝑔 𝑛 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑚 ∈𝑔 𝑛 ) ) → ∃ 𝑝 ∈ { 𝑦 ∈ V ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑦 = ( 𝑖 ∈𝑔 𝑗 ) } ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝 ⊼𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ) ) |
140 |
61 139
|
sylbi |
⊢ ( ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ∃ 𝑘 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖 ∈𝑔 𝑗 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑖 ∈𝑔 𝑗 ) ) → ∃ 𝑝 ∈ { 𝑦 ∈ V ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑦 = ( 𝑖 ∈𝑔 𝑗 ) } ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝 ⊼𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ) ) |
141 |
42 140
|
impbii |
⊢ ( ∃ 𝑝 ∈ { 𝑦 ∈ V ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑦 = ( 𝑖 ∈𝑔 𝑗 ) } ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝 ⊼𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ) ↔ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ∃ 𝑘 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖 ∈𝑔 𝑗 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑖 ∈𝑔 𝑗 ) ) ) |
142 |
8 141
|
bitri |
⊢ ( ∃ 𝑝 ∈ ( Fmla ‘ ∅ ) ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝 ⊼𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ) ↔ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ∃ 𝑘 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖 ∈𝑔 𝑗 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑖 ∈𝑔 𝑗 ) ) ) |
143 |
142
|
abbii |
⊢ { 𝑥 ∣ ∃ 𝑝 ∈ ( Fmla ‘ ∅ ) ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝 ⊼𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ) } = { 𝑥 ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ∃ 𝑘 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖 ∈𝑔 𝑗 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑖 ∈𝑔 𝑗 ) ) } |
144 |
6 143
|
uneq12i |
⊢ ( ( Fmla ‘ ∅ ) ∪ { 𝑥 ∣ ∃ 𝑝 ∈ ( Fmla ‘ ∅ ) ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝 ⊼𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ) } ) = ( ( { ∅ } × ( ω × ω ) ) ∪ { 𝑥 ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ∃ 𝑘 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖 ∈𝑔 𝑗 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑖 ∈𝑔 𝑗 ) ) } ) |
145 |
2 5 144
|
3eqtri |
⊢ ( Fmla ‘ 1o ) = ( ( { ∅ } × ( ω × ω ) ) ∪ { 𝑥 ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ∃ 𝑘 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖 ∈𝑔 𝑗 ) ⊼𝑔 ( 𝑘 ∈𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑖 ∈𝑔 𝑗 ) ) } ) |