Step |
Hyp |
Ref |
Expression |
1 |
|
peano2 |
⊢ ( 𝑁 ∈ ω → suc 𝑁 ∈ ω ) |
2 |
|
df-goal |
⊢ ∀𝑔 𝑖 𝑎 = 〈 2o , 〈 𝑖 , 𝑎 〉 〉 |
3 |
|
opex |
⊢ 〈 2o , 〈 𝑖 , 𝑎 〉 〉 ∈ V |
4 |
2 3
|
eqeltri |
⊢ ∀𝑔 𝑖 𝑎 ∈ V |
5 |
|
isfmlasuc |
⊢ ( ( suc 𝑁 ∈ ω ∧ ∀𝑔 𝑖 𝑎 ∈ V ) → ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ suc suc 𝑁 ) ↔ ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ suc 𝑁 ) ∨ ∃ 𝑢 ∈ ( Fmla ‘ suc 𝑁 ) ( ∃ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) ∀𝑔 𝑖 𝑎 = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑗 ∈ ω ∀𝑔 𝑖 𝑎 = ∀𝑔 𝑗 𝑢 ) ) ) ) |
6 |
1 4 5
|
sylancl |
⊢ ( 𝑁 ∈ ω → ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ suc suc 𝑁 ) ↔ ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ suc 𝑁 ) ∨ ∃ 𝑢 ∈ ( Fmla ‘ suc 𝑁 ) ( ∃ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) ∀𝑔 𝑖 𝑎 = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑗 ∈ ω ∀𝑔 𝑖 𝑎 = ∀𝑔 𝑗 𝑢 ) ) ) ) |
7 |
6
|
adantr |
⊢ ( ( 𝑁 ∈ ω ∧ ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ suc 𝑁 ) → 𝑎 ∈ ( Fmla ‘ suc 𝑁 ) ) ) → ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ suc suc 𝑁 ) ↔ ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ suc 𝑁 ) ∨ ∃ 𝑢 ∈ ( Fmla ‘ suc 𝑁 ) ( ∃ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) ∀𝑔 𝑖 𝑎 = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑗 ∈ ω ∀𝑔 𝑖 𝑎 = ∀𝑔 𝑗 𝑢 ) ) ) ) |
8 |
|
fmlasssuc |
⊢ ( suc 𝑁 ∈ ω → ( Fmla ‘ suc 𝑁 ) ⊆ ( Fmla ‘ suc suc 𝑁 ) ) |
9 |
1 8
|
syl |
⊢ ( 𝑁 ∈ ω → ( Fmla ‘ suc 𝑁 ) ⊆ ( Fmla ‘ suc suc 𝑁 ) ) |
10 |
9
|
sseld |
⊢ ( 𝑁 ∈ ω → ( 𝑎 ∈ ( Fmla ‘ suc 𝑁 ) → 𝑎 ∈ ( Fmla ‘ suc suc 𝑁 ) ) ) |
11 |
10
|
com12 |
⊢ ( 𝑎 ∈ ( Fmla ‘ suc 𝑁 ) → ( 𝑁 ∈ ω → 𝑎 ∈ ( Fmla ‘ suc suc 𝑁 ) ) ) |
12 |
11
|
imim2i |
⊢ ( ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ suc 𝑁 ) → 𝑎 ∈ ( Fmla ‘ suc 𝑁 ) ) → ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ suc 𝑁 ) → ( 𝑁 ∈ ω → 𝑎 ∈ ( Fmla ‘ suc suc 𝑁 ) ) ) ) |
13 |
12
|
com23 |
⊢ ( ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ suc 𝑁 ) → 𝑎 ∈ ( Fmla ‘ suc 𝑁 ) ) → ( 𝑁 ∈ ω → ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ suc 𝑁 ) → 𝑎 ∈ ( Fmla ‘ suc suc 𝑁 ) ) ) ) |
14 |
13
|
impcom |
⊢ ( ( 𝑁 ∈ ω ∧ ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ suc 𝑁 ) → 𝑎 ∈ ( Fmla ‘ suc 𝑁 ) ) ) → ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ suc 𝑁 ) → 𝑎 ∈ ( Fmla ‘ suc suc 𝑁 ) ) ) |
15 |
|
gonanegoal |
⊢ ( 𝑢 ⊼𝑔 𝑣 ) ≠ ∀𝑔 𝑖 𝑎 |
16 |
|
eqneqall |
⊢ ( ( 𝑢 ⊼𝑔 𝑣 ) = ∀𝑔 𝑖 𝑎 → ( ( 𝑢 ⊼𝑔 𝑣 ) ≠ ∀𝑔 𝑖 𝑎 → 𝑎 ∈ ( Fmla ‘ suc suc 𝑁 ) ) ) |
17 |
15 16
|
mpi |
⊢ ( ( 𝑢 ⊼𝑔 𝑣 ) = ∀𝑔 𝑖 𝑎 → 𝑎 ∈ ( Fmla ‘ suc suc 𝑁 ) ) |
18 |
17
|
eqcoms |
⊢ ( ∀𝑔 𝑖 𝑎 = ( 𝑢 ⊼𝑔 𝑣 ) → 𝑎 ∈ ( Fmla ‘ suc suc 𝑁 ) ) |
19 |
18
|
a1i |
⊢ ( ( ( 𝑁 ∈ ω ∧ 𝑢 ∈ ( Fmla ‘ suc 𝑁 ) ) ∧ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) ) → ( ∀𝑔 𝑖 𝑎 = ( 𝑢 ⊼𝑔 𝑣 ) → 𝑎 ∈ ( Fmla ‘ suc suc 𝑁 ) ) ) |
20 |
19
|
rexlimdva |
⊢ ( ( 𝑁 ∈ ω ∧ 𝑢 ∈ ( Fmla ‘ suc 𝑁 ) ) → ( ∃ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) ∀𝑔 𝑖 𝑎 = ( 𝑢 ⊼𝑔 𝑣 ) → 𝑎 ∈ ( Fmla ‘ suc suc 𝑁 ) ) ) |
21 |
|
df-goal |
⊢ ∀𝑔 𝑗 𝑢 = 〈 2o , 〈 𝑗 , 𝑢 〉 〉 |
22 |
2 21
|
eqeq12i |
⊢ ( ∀𝑔 𝑖 𝑎 = ∀𝑔 𝑗 𝑢 ↔ 〈 2o , 〈 𝑖 , 𝑎 〉 〉 = 〈 2o , 〈 𝑗 , 𝑢 〉 〉 ) |
23 |
|
2oex |
⊢ 2o ∈ V |
24 |
|
opex |
⊢ 〈 𝑖 , 𝑎 〉 ∈ V |
25 |
23 24
|
opth |
⊢ ( 〈 2o , 〈 𝑖 , 𝑎 〉 〉 = 〈 2o , 〈 𝑗 , 𝑢 〉 〉 ↔ ( 2o = 2o ∧ 〈 𝑖 , 𝑎 〉 = 〈 𝑗 , 𝑢 〉 ) ) |
26 |
22 25
|
bitri |
⊢ ( ∀𝑔 𝑖 𝑎 = ∀𝑔 𝑗 𝑢 ↔ ( 2o = 2o ∧ 〈 𝑖 , 𝑎 〉 = 〈 𝑗 , 𝑢 〉 ) ) |
27 |
|
vex |
⊢ 𝑖 ∈ V |
28 |
|
vex |
⊢ 𝑎 ∈ V |
29 |
27 28
|
opth |
⊢ ( 〈 𝑖 , 𝑎 〉 = 〈 𝑗 , 𝑢 〉 ↔ ( 𝑖 = 𝑗 ∧ 𝑎 = 𝑢 ) ) |
30 |
|
eleq1w |
⊢ ( 𝑢 = 𝑎 → ( 𝑢 ∈ ( Fmla ‘ suc 𝑁 ) ↔ 𝑎 ∈ ( Fmla ‘ suc 𝑁 ) ) ) |
31 |
30
|
eqcoms |
⊢ ( 𝑎 = 𝑢 → ( 𝑢 ∈ ( Fmla ‘ suc 𝑁 ) ↔ 𝑎 ∈ ( Fmla ‘ suc 𝑁 ) ) ) |
32 |
31 11
|
syl6bi |
⊢ ( 𝑎 = 𝑢 → ( 𝑢 ∈ ( Fmla ‘ suc 𝑁 ) → ( 𝑁 ∈ ω → 𝑎 ∈ ( Fmla ‘ suc suc 𝑁 ) ) ) ) |
33 |
32
|
impcomd |
⊢ ( 𝑎 = 𝑢 → ( ( 𝑁 ∈ ω ∧ 𝑢 ∈ ( Fmla ‘ suc 𝑁 ) ) → 𝑎 ∈ ( Fmla ‘ suc suc 𝑁 ) ) ) |
34 |
29 33
|
simplbiim |
⊢ ( 〈 𝑖 , 𝑎 〉 = 〈 𝑗 , 𝑢 〉 → ( ( 𝑁 ∈ ω ∧ 𝑢 ∈ ( Fmla ‘ suc 𝑁 ) ) → 𝑎 ∈ ( Fmla ‘ suc suc 𝑁 ) ) ) |
35 |
26 34
|
simplbiim |
⊢ ( ∀𝑔 𝑖 𝑎 = ∀𝑔 𝑗 𝑢 → ( ( 𝑁 ∈ ω ∧ 𝑢 ∈ ( Fmla ‘ suc 𝑁 ) ) → 𝑎 ∈ ( Fmla ‘ suc suc 𝑁 ) ) ) |
36 |
35
|
com12 |
⊢ ( ( 𝑁 ∈ ω ∧ 𝑢 ∈ ( Fmla ‘ suc 𝑁 ) ) → ( ∀𝑔 𝑖 𝑎 = ∀𝑔 𝑗 𝑢 → 𝑎 ∈ ( Fmla ‘ suc suc 𝑁 ) ) ) |
37 |
36
|
adantr |
⊢ ( ( ( 𝑁 ∈ ω ∧ 𝑢 ∈ ( Fmla ‘ suc 𝑁 ) ) ∧ 𝑗 ∈ ω ) → ( ∀𝑔 𝑖 𝑎 = ∀𝑔 𝑗 𝑢 → 𝑎 ∈ ( Fmla ‘ suc suc 𝑁 ) ) ) |
38 |
37
|
rexlimdva |
⊢ ( ( 𝑁 ∈ ω ∧ 𝑢 ∈ ( Fmla ‘ suc 𝑁 ) ) → ( ∃ 𝑗 ∈ ω ∀𝑔 𝑖 𝑎 = ∀𝑔 𝑗 𝑢 → 𝑎 ∈ ( Fmla ‘ suc suc 𝑁 ) ) ) |
39 |
20 38
|
jaod |
⊢ ( ( 𝑁 ∈ ω ∧ 𝑢 ∈ ( Fmla ‘ suc 𝑁 ) ) → ( ( ∃ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) ∀𝑔 𝑖 𝑎 = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑗 ∈ ω ∀𝑔 𝑖 𝑎 = ∀𝑔 𝑗 𝑢 ) → 𝑎 ∈ ( Fmla ‘ suc suc 𝑁 ) ) ) |
40 |
39
|
rexlimdva |
⊢ ( 𝑁 ∈ ω → ( ∃ 𝑢 ∈ ( Fmla ‘ suc 𝑁 ) ( ∃ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) ∀𝑔 𝑖 𝑎 = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑗 ∈ ω ∀𝑔 𝑖 𝑎 = ∀𝑔 𝑗 𝑢 ) → 𝑎 ∈ ( Fmla ‘ suc suc 𝑁 ) ) ) |
41 |
40
|
adantr |
⊢ ( ( 𝑁 ∈ ω ∧ ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ suc 𝑁 ) → 𝑎 ∈ ( Fmla ‘ suc 𝑁 ) ) ) → ( ∃ 𝑢 ∈ ( Fmla ‘ suc 𝑁 ) ( ∃ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) ∀𝑔 𝑖 𝑎 = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑗 ∈ ω ∀𝑔 𝑖 𝑎 = ∀𝑔 𝑗 𝑢 ) → 𝑎 ∈ ( Fmla ‘ suc suc 𝑁 ) ) ) |
42 |
14 41
|
jaod |
⊢ ( ( 𝑁 ∈ ω ∧ ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ suc 𝑁 ) → 𝑎 ∈ ( Fmla ‘ suc 𝑁 ) ) ) → ( ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ suc 𝑁 ) ∨ ∃ 𝑢 ∈ ( Fmla ‘ suc 𝑁 ) ( ∃ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) ∀𝑔 𝑖 𝑎 = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑗 ∈ ω ∀𝑔 𝑖 𝑎 = ∀𝑔 𝑗 𝑢 ) ) → 𝑎 ∈ ( Fmla ‘ suc suc 𝑁 ) ) ) |
43 |
7 42
|
sylbid |
⊢ ( ( 𝑁 ∈ ω ∧ ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ suc 𝑁 ) → 𝑎 ∈ ( Fmla ‘ suc 𝑁 ) ) ) → ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ suc suc 𝑁 ) → 𝑎 ∈ ( Fmla ‘ suc suc 𝑁 ) ) ) |
44 |
43
|
ex |
⊢ ( 𝑁 ∈ ω → ( ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ suc 𝑁 ) → 𝑎 ∈ ( Fmla ‘ suc 𝑁 ) ) → ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ suc suc 𝑁 ) → 𝑎 ∈ ( Fmla ‘ suc suc 𝑁 ) ) ) ) |