Step |
Hyp |
Ref |
Expression |
1 |
|
peano2 |
⊢ ( 𝑁 ∈ ω → suc 𝑁 ∈ ω ) |
2 |
|
ovexd |
⊢ ( 𝑁 ∈ ω → ( 𝑎 ⊼𝑔 𝑏 ) ∈ V ) |
3 |
|
isfmlasuc |
⊢ ( ( suc 𝑁 ∈ ω ∧ ( 𝑎 ⊼𝑔 𝑏 ) ∈ V ) → ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ suc suc 𝑁 ) ↔ ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ suc 𝑁 ) ∨ ∃ 𝑢 ∈ ( Fmla ‘ suc 𝑁 ) ( ∃ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) ( 𝑎 ⊼𝑔 𝑏 ) = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω ( 𝑎 ⊼𝑔 𝑏 ) = ∀𝑔 𝑖 𝑢 ) ) ) ) |
4 |
1 2 3
|
syl2anc |
⊢ ( 𝑁 ∈ ω → ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ suc suc 𝑁 ) ↔ ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ suc 𝑁 ) ∨ ∃ 𝑢 ∈ ( Fmla ‘ suc 𝑁 ) ( ∃ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) ( 𝑎 ⊼𝑔 𝑏 ) = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω ( 𝑎 ⊼𝑔 𝑏 ) = ∀𝑔 𝑖 𝑢 ) ) ) ) |
5 |
4
|
adantr |
⊢ ( ( 𝑁 ∈ ω ∧ ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ suc 𝑁 ) → ( 𝑎 ∈ ( Fmla ‘ suc 𝑁 ) ∧ 𝑏 ∈ ( Fmla ‘ suc 𝑁 ) ) ) ) → ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ suc suc 𝑁 ) ↔ ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ suc 𝑁 ) ∨ ∃ 𝑢 ∈ ( Fmla ‘ suc 𝑁 ) ( ∃ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) ( 𝑎 ⊼𝑔 𝑏 ) = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω ( 𝑎 ⊼𝑔 𝑏 ) = ∀𝑔 𝑖 𝑢 ) ) ) ) |
6 |
|
fmlasssuc |
⊢ ( suc 𝑁 ∈ ω → ( Fmla ‘ suc 𝑁 ) ⊆ ( Fmla ‘ suc suc 𝑁 ) ) |
7 |
1 6
|
syl |
⊢ ( 𝑁 ∈ ω → ( Fmla ‘ suc 𝑁 ) ⊆ ( Fmla ‘ suc suc 𝑁 ) ) |
8 |
7
|
sseld |
⊢ ( 𝑁 ∈ ω → ( 𝑎 ∈ ( Fmla ‘ suc 𝑁 ) → 𝑎 ∈ ( Fmla ‘ suc suc 𝑁 ) ) ) |
9 |
7
|
sseld |
⊢ ( 𝑁 ∈ ω → ( 𝑏 ∈ ( Fmla ‘ suc 𝑁 ) → 𝑏 ∈ ( Fmla ‘ suc suc 𝑁 ) ) ) |
10 |
8 9
|
anim12d |
⊢ ( 𝑁 ∈ ω → ( ( 𝑎 ∈ ( Fmla ‘ suc 𝑁 ) ∧ 𝑏 ∈ ( Fmla ‘ suc 𝑁 ) ) → ( 𝑎 ∈ ( Fmla ‘ suc suc 𝑁 ) ∧ 𝑏 ∈ ( Fmla ‘ suc suc 𝑁 ) ) ) ) |
11 |
10
|
com12 |
⊢ ( ( 𝑎 ∈ ( Fmla ‘ suc 𝑁 ) ∧ 𝑏 ∈ ( Fmla ‘ suc 𝑁 ) ) → ( 𝑁 ∈ ω → ( 𝑎 ∈ ( Fmla ‘ suc suc 𝑁 ) ∧ 𝑏 ∈ ( Fmla ‘ suc suc 𝑁 ) ) ) ) |
12 |
11
|
imim2i |
⊢ ( ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ suc 𝑁 ) → ( 𝑎 ∈ ( Fmla ‘ suc 𝑁 ) ∧ 𝑏 ∈ ( Fmla ‘ suc 𝑁 ) ) ) → ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ suc 𝑁 ) → ( 𝑁 ∈ ω → ( 𝑎 ∈ ( Fmla ‘ suc suc 𝑁 ) ∧ 𝑏 ∈ ( Fmla ‘ suc suc 𝑁 ) ) ) ) ) |
13 |
12
|
com23 |
⊢ ( ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ suc 𝑁 ) → ( 𝑎 ∈ ( Fmla ‘ suc 𝑁 ) ∧ 𝑏 ∈ ( Fmla ‘ suc 𝑁 ) ) ) → ( 𝑁 ∈ ω → ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ suc 𝑁 ) → ( 𝑎 ∈ ( Fmla ‘ suc suc 𝑁 ) ∧ 𝑏 ∈ ( Fmla ‘ suc suc 𝑁 ) ) ) ) ) |
14 |
13
|
impcom |
⊢ ( ( 𝑁 ∈ ω ∧ ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ suc 𝑁 ) → ( 𝑎 ∈ ( Fmla ‘ suc 𝑁 ) ∧ 𝑏 ∈ ( Fmla ‘ suc 𝑁 ) ) ) ) → ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ suc 𝑁 ) → ( 𝑎 ∈ ( Fmla ‘ suc suc 𝑁 ) ∧ 𝑏 ∈ ( Fmla ‘ suc suc 𝑁 ) ) ) ) |
15 |
|
gonafv |
⊢ ( ( 𝑎 ∈ V ∧ 𝑏 ∈ V ) → ( 𝑎 ⊼𝑔 𝑏 ) = 〈 1o , 〈 𝑎 , 𝑏 〉 〉 ) |
16 |
15
|
el2v |
⊢ ( 𝑎 ⊼𝑔 𝑏 ) = 〈 1o , 〈 𝑎 , 𝑏 〉 〉 |
17 |
16
|
a1i |
⊢ ( ( 𝑢 ∈ ( Fmla ‘ suc 𝑁 ) ∧ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) ) → ( 𝑎 ⊼𝑔 𝑏 ) = 〈 1o , 〈 𝑎 , 𝑏 〉 〉 ) |
18 |
|
gonafv |
⊢ ( ( 𝑢 ∈ ( Fmla ‘ suc 𝑁 ) ∧ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) ) → ( 𝑢 ⊼𝑔 𝑣 ) = 〈 1o , 〈 𝑢 , 𝑣 〉 〉 ) |
19 |
17 18
|
eqeq12d |
⊢ ( ( 𝑢 ∈ ( Fmla ‘ suc 𝑁 ) ∧ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) ) → ( ( 𝑎 ⊼𝑔 𝑏 ) = ( 𝑢 ⊼𝑔 𝑣 ) ↔ 〈 1o , 〈 𝑎 , 𝑏 〉 〉 = 〈 1o , 〈 𝑢 , 𝑣 〉 〉 ) ) |
20 |
|
1oex |
⊢ 1o ∈ V |
21 |
|
opex |
⊢ 〈 𝑎 , 𝑏 〉 ∈ V |
22 |
20 21
|
opth |
⊢ ( 〈 1o , 〈 𝑎 , 𝑏 〉 〉 = 〈 1o , 〈 𝑢 , 𝑣 〉 〉 ↔ ( 1o = 1o ∧ 〈 𝑎 , 𝑏 〉 = 〈 𝑢 , 𝑣 〉 ) ) |
23 |
19 22
|
syl6bb |
⊢ ( ( 𝑢 ∈ ( Fmla ‘ suc 𝑁 ) ∧ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) ) → ( ( 𝑎 ⊼𝑔 𝑏 ) = ( 𝑢 ⊼𝑔 𝑣 ) ↔ ( 1o = 1o ∧ 〈 𝑎 , 𝑏 〉 = 〈 𝑢 , 𝑣 〉 ) ) ) |
24 |
23
|
adantll |
⊢ ( ( ( 𝑁 ∈ ω ∧ 𝑢 ∈ ( Fmla ‘ suc 𝑁 ) ) ∧ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) ) → ( ( 𝑎 ⊼𝑔 𝑏 ) = ( 𝑢 ⊼𝑔 𝑣 ) ↔ ( 1o = 1o ∧ 〈 𝑎 , 𝑏 〉 = 〈 𝑢 , 𝑣 〉 ) ) ) |
25 |
|
vex |
⊢ 𝑎 ∈ V |
26 |
|
vex |
⊢ 𝑏 ∈ V |
27 |
25 26
|
opth |
⊢ ( 〈 𝑎 , 𝑏 〉 = 〈 𝑢 , 𝑣 〉 ↔ ( 𝑎 = 𝑢 ∧ 𝑏 = 𝑣 ) ) |
28 |
|
eleq1w |
⊢ ( 𝑢 = 𝑎 → ( 𝑢 ∈ ( Fmla ‘ suc 𝑁 ) ↔ 𝑎 ∈ ( Fmla ‘ suc 𝑁 ) ) ) |
29 |
28
|
equcoms |
⊢ ( 𝑎 = 𝑢 → ( 𝑢 ∈ ( Fmla ‘ suc 𝑁 ) ↔ 𝑎 ∈ ( Fmla ‘ suc 𝑁 ) ) ) |
30 |
|
eleq1w |
⊢ ( 𝑣 = 𝑏 → ( 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) ↔ 𝑏 ∈ ( Fmla ‘ suc 𝑁 ) ) ) |
31 |
30
|
equcoms |
⊢ ( 𝑏 = 𝑣 → ( 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) ↔ 𝑏 ∈ ( Fmla ‘ suc 𝑁 ) ) ) |
32 |
29 31
|
bi2anan9 |
⊢ ( ( 𝑎 = 𝑢 ∧ 𝑏 = 𝑣 ) → ( ( 𝑢 ∈ ( Fmla ‘ suc 𝑁 ) ∧ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) ) ↔ ( 𝑎 ∈ ( Fmla ‘ suc 𝑁 ) ∧ 𝑏 ∈ ( Fmla ‘ suc 𝑁 ) ) ) ) |
33 |
32 11
|
syl6bi |
⊢ ( ( 𝑎 = 𝑢 ∧ 𝑏 = 𝑣 ) → ( ( 𝑢 ∈ ( Fmla ‘ suc 𝑁 ) ∧ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) ) → ( 𝑁 ∈ ω → ( 𝑎 ∈ ( Fmla ‘ suc suc 𝑁 ) ∧ 𝑏 ∈ ( Fmla ‘ suc suc 𝑁 ) ) ) ) ) |
34 |
27 33
|
sylbi |
⊢ ( 〈 𝑎 , 𝑏 〉 = 〈 𝑢 , 𝑣 〉 → ( ( 𝑢 ∈ ( Fmla ‘ suc 𝑁 ) ∧ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) ) → ( 𝑁 ∈ ω → ( 𝑎 ∈ ( Fmla ‘ suc suc 𝑁 ) ∧ 𝑏 ∈ ( Fmla ‘ suc suc 𝑁 ) ) ) ) ) |
35 |
34
|
adantl |
⊢ ( ( 1o = 1o ∧ 〈 𝑎 , 𝑏 〉 = 〈 𝑢 , 𝑣 〉 ) → ( ( 𝑢 ∈ ( Fmla ‘ suc 𝑁 ) ∧ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) ) → ( 𝑁 ∈ ω → ( 𝑎 ∈ ( Fmla ‘ suc suc 𝑁 ) ∧ 𝑏 ∈ ( Fmla ‘ suc suc 𝑁 ) ) ) ) ) |
36 |
35
|
com13 |
⊢ ( 𝑁 ∈ ω → ( ( 𝑢 ∈ ( Fmla ‘ suc 𝑁 ) ∧ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) ) → ( ( 1o = 1o ∧ 〈 𝑎 , 𝑏 〉 = 〈 𝑢 , 𝑣 〉 ) → ( 𝑎 ∈ ( Fmla ‘ suc suc 𝑁 ) ∧ 𝑏 ∈ ( Fmla ‘ suc suc 𝑁 ) ) ) ) ) |
37 |
36
|
impl |
⊢ ( ( ( 𝑁 ∈ ω ∧ 𝑢 ∈ ( Fmla ‘ suc 𝑁 ) ) ∧ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) ) → ( ( 1o = 1o ∧ 〈 𝑎 , 𝑏 〉 = 〈 𝑢 , 𝑣 〉 ) → ( 𝑎 ∈ ( Fmla ‘ suc suc 𝑁 ) ∧ 𝑏 ∈ ( Fmla ‘ suc suc 𝑁 ) ) ) ) |
38 |
24 37
|
sylbid |
⊢ ( ( ( 𝑁 ∈ ω ∧ 𝑢 ∈ ( Fmla ‘ suc 𝑁 ) ) ∧ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) ) → ( ( 𝑎 ⊼𝑔 𝑏 ) = ( 𝑢 ⊼𝑔 𝑣 ) → ( 𝑎 ∈ ( Fmla ‘ suc suc 𝑁 ) ∧ 𝑏 ∈ ( Fmla ‘ suc suc 𝑁 ) ) ) ) |
39 |
38
|
rexlimdva |
⊢ ( ( 𝑁 ∈ ω ∧ 𝑢 ∈ ( Fmla ‘ suc 𝑁 ) ) → ( ∃ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) ( 𝑎 ⊼𝑔 𝑏 ) = ( 𝑢 ⊼𝑔 𝑣 ) → ( 𝑎 ∈ ( Fmla ‘ suc suc 𝑁 ) ∧ 𝑏 ∈ ( Fmla ‘ suc suc 𝑁 ) ) ) ) |
40 |
|
gonanegoal |
⊢ ( 𝑎 ⊼𝑔 𝑏 ) ≠ ∀𝑔 𝑖 𝑢 |
41 |
|
eqneqall |
⊢ ( ( 𝑎 ⊼𝑔 𝑏 ) = ∀𝑔 𝑖 𝑢 → ( ( 𝑎 ⊼𝑔 𝑏 ) ≠ ∀𝑔 𝑖 𝑢 → ( 𝑎 ∈ ( Fmla ‘ suc suc 𝑁 ) ∧ 𝑏 ∈ ( Fmla ‘ suc suc 𝑁 ) ) ) ) |
42 |
40 41
|
mpi |
⊢ ( ( 𝑎 ⊼𝑔 𝑏 ) = ∀𝑔 𝑖 𝑢 → ( 𝑎 ∈ ( Fmla ‘ suc suc 𝑁 ) ∧ 𝑏 ∈ ( Fmla ‘ suc suc 𝑁 ) ) ) |
43 |
42
|
a1i |
⊢ ( ( ( 𝑁 ∈ ω ∧ 𝑢 ∈ ( Fmla ‘ suc 𝑁 ) ) ∧ 𝑖 ∈ ω ) → ( ( 𝑎 ⊼𝑔 𝑏 ) = ∀𝑔 𝑖 𝑢 → ( 𝑎 ∈ ( Fmla ‘ suc suc 𝑁 ) ∧ 𝑏 ∈ ( Fmla ‘ suc suc 𝑁 ) ) ) ) |
44 |
43
|
rexlimdva |
⊢ ( ( 𝑁 ∈ ω ∧ 𝑢 ∈ ( Fmla ‘ suc 𝑁 ) ) → ( ∃ 𝑖 ∈ ω ( 𝑎 ⊼𝑔 𝑏 ) = ∀𝑔 𝑖 𝑢 → ( 𝑎 ∈ ( Fmla ‘ suc suc 𝑁 ) ∧ 𝑏 ∈ ( Fmla ‘ suc suc 𝑁 ) ) ) ) |
45 |
39 44
|
jaod |
⊢ ( ( 𝑁 ∈ ω ∧ 𝑢 ∈ ( Fmla ‘ suc 𝑁 ) ) → ( ( ∃ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) ( 𝑎 ⊼𝑔 𝑏 ) = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω ( 𝑎 ⊼𝑔 𝑏 ) = ∀𝑔 𝑖 𝑢 ) → ( 𝑎 ∈ ( Fmla ‘ suc suc 𝑁 ) ∧ 𝑏 ∈ ( Fmla ‘ suc suc 𝑁 ) ) ) ) |
46 |
45
|
rexlimdva |
⊢ ( 𝑁 ∈ ω → ( ∃ 𝑢 ∈ ( Fmla ‘ suc 𝑁 ) ( ∃ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) ( 𝑎 ⊼𝑔 𝑏 ) = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω ( 𝑎 ⊼𝑔 𝑏 ) = ∀𝑔 𝑖 𝑢 ) → ( 𝑎 ∈ ( Fmla ‘ suc suc 𝑁 ) ∧ 𝑏 ∈ ( Fmla ‘ suc suc 𝑁 ) ) ) ) |
47 |
46
|
adantr |
⊢ ( ( 𝑁 ∈ ω ∧ ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ suc 𝑁 ) → ( 𝑎 ∈ ( Fmla ‘ suc 𝑁 ) ∧ 𝑏 ∈ ( Fmla ‘ suc 𝑁 ) ) ) ) → ( ∃ 𝑢 ∈ ( Fmla ‘ suc 𝑁 ) ( ∃ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) ( 𝑎 ⊼𝑔 𝑏 ) = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω ( 𝑎 ⊼𝑔 𝑏 ) = ∀𝑔 𝑖 𝑢 ) → ( 𝑎 ∈ ( Fmla ‘ suc suc 𝑁 ) ∧ 𝑏 ∈ ( Fmla ‘ suc suc 𝑁 ) ) ) ) |
48 |
14 47
|
jaod |
⊢ ( ( 𝑁 ∈ ω ∧ ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ suc 𝑁 ) → ( 𝑎 ∈ ( Fmla ‘ suc 𝑁 ) ∧ 𝑏 ∈ ( Fmla ‘ suc 𝑁 ) ) ) ) → ( ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ suc 𝑁 ) ∨ ∃ 𝑢 ∈ ( Fmla ‘ suc 𝑁 ) ( ∃ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) ( 𝑎 ⊼𝑔 𝑏 ) = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω ( 𝑎 ⊼𝑔 𝑏 ) = ∀𝑔 𝑖 𝑢 ) ) → ( 𝑎 ∈ ( Fmla ‘ suc suc 𝑁 ) ∧ 𝑏 ∈ ( Fmla ‘ suc suc 𝑁 ) ) ) ) |
49 |
5 48
|
sylbid |
⊢ ( ( 𝑁 ∈ ω ∧ ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ suc 𝑁 ) → ( 𝑎 ∈ ( Fmla ‘ suc 𝑁 ) ∧ 𝑏 ∈ ( Fmla ‘ suc 𝑁 ) ) ) ) → ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ suc suc 𝑁 ) → ( 𝑎 ∈ ( Fmla ‘ suc suc 𝑁 ) ∧ 𝑏 ∈ ( Fmla ‘ suc suc 𝑁 ) ) ) ) |
50 |
49
|
ex |
⊢ ( 𝑁 ∈ ω → ( ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ suc 𝑁 ) → ( 𝑎 ∈ ( Fmla ‘ suc 𝑁 ) ∧ 𝑏 ∈ ( Fmla ‘ suc 𝑁 ) ) ) → ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ suc suc 𝑁 ) → ( 𝑎 ∈ ( Fmla ‘ suc suc 𝑁 ) ∧ 𝑏 ∈ ( Fmla ‘ suc suc 𝑁 ) ) ) ) ) |