Step |
Hyp |
Ref |
Expression |
1 |
|
vex |
⊢ 𝑓 ∈ V |
2 |
|
eqeq1 |
⊢ ( 𝑥 = 𝑓 → ( 𝑥 = ( 𝑢 ⊼𝑔 𝑣 ) ↔ 𝑓 = ( 𝑢 ⊼𝑔 𝑣 ) ) ) |
3 |
2
|
rexbidv |
⊢ ( 𝑥 = 𝑓 → ( ∃ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑢 ⊼𝑔 𝑣 ) ↔ ∃ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) 𝑓 = ( 𝑢 ⊼𝑔 𝑣 ) ) ) |
4 |
|
eqeq1 |
⊢ ( 𝑥 = 𝑓 → ( 𝑥 = ∀𝑔 𝑖 𝑢 ↔ 𝑓 = ∀𝑔 𝑖 𝑢 ) ) |
5 |
4
|
rexbidv |
⊢ ( 𝑥 = 𝑓 → ( ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑢 ↔ ∃ 𝑖 ∈ ω 𝑓 = ∀𝑔 𝑖 𝑢 ) ) |
6 |
3 5
|
orbi12d |
⊢ ( 𝑥 = 𝑓 → ( ( ∃ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑢 ) ↔ ( ∃ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) 𝑓 = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω 𝑓 = ∀𝑔 𝑖 𝑢 ) ) ) |
7 |
6
|
rexbidv |
⊢ ( 𝑥 = 𝑓 → ( ∃ 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑢 ) ↔ ∃ 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) 𝑓 = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω 𝑓 = ∀𝑔 𝑖 𝑢 ) ) ) |
8 |
2
|
2rexbidv |
⊢ ( 𝑥 = 𝑓 → ( ∃ 𝑢 ∈ ( Fmla ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑢 ⊼𝑔 𝑣 ) ↔ ∃ 𝑢 ∈ ( Fmla ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑓 = ( 𝑢 ⊼𝑔 𝑣 ) ) ) |
9 |
7 8
|
orbi12d |
⊢ ( 𝑥 = 𝑓 → ( ( ∃ 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑢 ) ∨ ∃ 𝑢 ∈ ( Fmla ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑢 ⊼𝑔 𝑣 ) ) ↔ ( ∃ 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) 𝑓 = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω 𝑓 = ∀𝑔 𝑖 𝑢 ) ∨ ∃ 𝑢 ∈ ( Fmla ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑓 = ( 𝑢 ⊼𝑔 𝑣 ) ) ) ) |
10 |
1 9
|
elab |
⊢ ( 𝑓 ∈ { 𝑥 ∣ ( ∃ 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑢 ) ∨ ∃ 𝑢 ∈ ( Fmla ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑢 ⊼𝑔 𝑣 ) ) } ↔ ( ∃ 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) 𝑓 = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω 𝑓 = ∀𝑔 𝑖 𝑢 ) ∨ ∃ 𝑢 ∈ ( Fmla ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑓 = ( 𝑢 ⊼𝑔 𝑣 ) ) ) |
11 |
|
gonar |
⊢ ( ( 𝑁 ∈ ω ∧ ( 𝑢 ⊼𝑔 𝑣 ) ∈ ( Fmla ‘ 𝑁 ) ) → ( 𝑢 ∈ ( Fmla ‘ 𝑁 ) ∧ 𝑣 ∈ ( Fmla ‘ 𝑁 ) ) ) |
12 |
|
elndif |
⊢ ( 𝑢 ∈ ( Fmla ‘ 𝑁 ) → ¬ 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) |
13 |
12
|
adantr |
⊢ ( ( 𝑢 ∈ ( Fmla ‘ 𝑁 ) ∧ 𝑣 ∈ ( Fmla ‘ 𝑁 ) ) → ¬ 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) |
14 |
13
|
intnanrd |
⊢ ( ( 𝑢 ∈ ( Fmla ‘ 𝑁 ) ∧ 𝑣 ∈ ( Fmla ‘ 𝑁 ) ) → ¬ ( 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ∧ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) ) ) |
15 |
11 14
|
syl |
⊢ ( ( 𝑁 ∈ ω ∧ ( 𝑢 ⊼𝑔 𝑣 ) ∈ ( Fmla ‘ 𝑁 ) ) → ¬ ( 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ∧ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) ) ) |
16 |
15
|
ex |
⊢ ( 𝑁 ∈ ω → ( ( 𝑢 ⊼𝑔 𝑣 ) ∈ ( Fmla ‘ 𝑁 ) → ¬ ( 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ∧ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) ) ) ) |
17 |
16
|
con2d |
⊢ ( 𝑁 ∈ ω → ( ( 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ∧ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) ) → ¬ ( 𝑢 ⊼𝑔 𝑣 ) ∈ ( Fmla ‘ 𝑁 ) ) ) |
18 |
17
|
impl |
⊢ ( ( ( 𝑁 ∈ ω ∧ 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) ∧ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) ) → ¬ ( 𝑢 ⊼𝑔 𝑣 ) ∈ ( Fmla ‘ 𝑁 ) ) |
19 |
|
elneeldif |
⊢ ( ( 𝑎 ∈ ( Fmla ‘ 𝑁 ) ∧ 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) → 𝑎 ≠ 𝑢 ) |
20 |
19
|
necomd |
⊢ ( ( 𝑎 ∈ ( Fmla ‘ 𝑁 ) ∧ 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) → 𝑢 ≠ 𝑎 ) |
21 |
20
|
ancoms |
⊢ ( ( 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ∧ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ) → 𝑢 ≠ 𝑎 ) |
22 |
21
|
neneqd |
⊢ ( ( 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ∧ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ) → ¬ 𝑢 = 𝑎 ) |
23 |
22
|
orcd |
⊢ ( ( 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ∧ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ) → ( ¬ 𝑢 = 𝑎 ∨ ¬ 𝑣 = 𝑏 ) ) |
24 |
|
ianor |
⊢ ( ¬ ( 𝑢 = 𝑎 ∧ 𝑣 = 𝑏 ) ↔ ( ¬ 𝑢 = 𝑎 ∨ ¬ 𝑣 = 𝑏 ) ) |
25 |
|
vex |
⊢ 𝑢 ∈ V |
26 |
|
vex |
⊢ 𝑣 ∈ V |
27 |
25 26
|
opth |
⊢ ( 〈 𝑢 , 𝑣 〉 = 〈 𝑎 , 𝑏 〉 ↔ ( 𝑢 = 𝑎 ∧ 𝑣 = 𝑏 ) ) |
28 |
24 27
|
xchnxbir |
⊢ ( ¬ 〈 𝑢 , 𝑣 〉 = 〈 𝑎 , 𝑏 〉 ↔ ( ¬ 𝑢 = 𝑎 ∨ ¬ 𝑣 = 𝑏 ) ) |
29 |
23 28
|
sylibr |
⊢ ( ( 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ∧ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ) → ¬ 〈 𝑢 , 𝑣 〉 = 〈 𝑎 , 𝑏 〉 ) |
30 |
29
|
olcd |
⊢ ( ( 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ∧ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ) → ( ¬ 1o = 1o ∨ ¬ 〈 𝑢 , 𝑣 〉 = 〈 𝑎 , 𝑏 〉 ) ) |
31 |
|
ianor |
⊢ ( ¬ ( 1o = 1o ∧ 〈 𝑢 , 𝑣 〉 = 〈 𝑎 , 𝑏 〉 ) ↔ ( ¬ 1o = 1o ∨ ¬ 〈 𝑢 , 𝑣 〉 = 〈 𝑎 , 𝑏 〉 ) ) |
32 |
|
gonafv |
⊢ ( ( 𝑢 ∈ V ∧ 𝑣 ∈ V ) → ( 𝑢 ⊼𝑔 𝑣 ) = 〈 1o , 〈 𝑢 , 𝑣 〉 〉 ) |
33 |
32
|
el2v |
⊢ ( 𝑢 ⊼𝑔 𝑣 ) = 〈 1o , 〈 𝑢 , 𝑣 〉 〉 |
34 |
|
gonafv |
⊢ ( ( 𝑎 ∈ V ∧ 𝑏 ∈ V ) → ( 𝑎 ⊼𝑔 𝑏 ) = 〈 1o , 〈 𝑎 , 𝑏 〉 〉 ) |
35 |
34
|
el2v |
⊢ ( 𝑎 ⊼𝑔 𝑏 ) = 〈 1o , 〈 𝑎 , 𝑏 〉 〉 |
36 |
33 35
|
eqeq12i |
⊢ ( ( 𝑢 ⊼𝑔 𝑣 ) = ( 𝑎 ⊼𝑔 𝑏 ) ↔ 〈 1o , 〈 𝑢 , 𝑣 〉 〉 = 〈 1o , 〈 𝑎 , 𝑏 〉 〉 ) |
37 |
|
1oex |
⊢ 1o ∈ V |
38 |
|
opex |
⊢ 〈 𝑢 , 𝑣 〉 ∈ V |
39 |
37 38
|
opth |
⊢ ( 〈 1o , 〈 𝑢 , 𝑣 〉 〉 = 〈 1o , 〈 𝑎 , 𝑏 〉 〉 ↔ ( 1o = 1o ∧ 〈 𝑢 , 𝑣 〉 = 〈 𝑎 , 𝑏 〉 ) ) |
40 |
36 39
|
bitri |
⊢ ( ( 𝑢 ⊼𝑔 𝑣 ) = ( 𝑎 ⊼𝑔 𝑏 ) ↔ ( 1o = 1o ∧ 〈 𝑢 , 𝑣 〉 = 〈 𝑎 , 𝑏 〉 ) ) |
41 |
31 40
|
xchnxbir |
⊢ ( ¬ ( 𝑢 ⊼𝑔 𝑣 ) = ( 𝑎 ⊼𝑔 𝑏 ) ↔ ( ¬ 1o = 1o ∨ ¬ 〈 𝑢 , 𝑣 〉 = 〈 𝑎 , 𝑏 〉 ) ) |
42 |
30 41
|
sylibr |
⊢ ( ( 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ∧ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ) → ¬ ( 𝑢 ⊼𝑔 𝑣 ) = ( 𝑎 ⊼𝑔 𝑏 ) ) |
43 |
42
|
ralrimivw |
⊢ ( ( 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ∧ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ) → ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ ( 𝑢 ⊼𝑔 𝑣 ) = ( 𝑎 ⊼𝑔 𝑏 ) ) |
44 |
43
|
ralrimiva |
⊢ ( 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) → ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ ( 𝑢 ⊼𝑔 𝑣 ) = ( 𝑎 ⊼𝑔 𝑏 ) ) |
45 |
44
|
adantl |
⊢ ( ( 𝑁 ∈ ω ∧ 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) → ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ ( 𝑢 ⊼𝑔 𝑣 ) = ( 𝑎 ⊼𝑔 𝑏 ) ) |
46 |
45
|
adantr |
⊢ ( ( ( 𝑁 ∈ ω ∧ 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) ∧ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) ) → ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ ( 𝑢 ⊼𝑔 𝑣 ) = ( 𝑎 ⊼𝑔 𝑏 ) ) |
47 |
|
gonanegoal |
⊢ ( 𝑢 ⊼𝑔 𝑣 ) ≠ ∀𝑔 𝑗 𝑎 |
48 |
47
|
neii |
⊢ ¬ ( 𝑢 ⊼𝑔 𝑣 ) = ∀𝑔 𝑗 𝑎 |
49 |
48
|
a1i |
⊢ ( ( ( 𝑁 ∈ ω ∧ 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) ∧ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) ) → ¬ ( 𝑢 ⊼𝑔 𝑣 ) = ∀𝑔 𝑗 𝑎 ) |
50 |
49
|
ralrimivw |
⊢ ( ( ( 𝑁 ∈ ω ∧ 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) ∧ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) ) → ∀ 𝑗 ∈ ω ¬ ( 𝑢 ⊼𝑔 𝑣 ) = ∀𝑔 𝑗 𝑎 ) |
51 |
50
|
ralrimivw |
⊢ ( ( ( 𝑁 ∈ ω ∧ 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) ∧ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) ) → ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ∀ 𝑗 ∈ ω ¬ ( 𝑢 ⊼𝑔 𝑣 ) = ∀𝑔 𝑗 𝑎 ) |
52 |
|
r19.26 |
⊢ ( ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ( ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ ( 𝑢 ⊼𝑔 𝑣 ) = ( 𝑎 ⊼𝑔 𝑏 ) ∧ ∀ 𝑗 ∈ ω ¬ ( 𝑢 ⊼𝑔 𝑣 ) = ∀𝑔 𝑗 𝑎 ) ↔ ( ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ ( 𝑢 ⊼𝑔 𝑣 ) = ( 𝑎 ⊼𝑔 𝑏 ) ∧ ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ∀ 𝑗 ∈ ω ¬ ( 𝑢 ⊼𝑔 𝑣 ) = ∀𝑔 𝑗 𝑎 ) ) |
53 |
46 51 52
|
sylanbrc |
⊢ ( ( ( 𝑁 ∈ ω ∧ 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) ∧ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) ) → ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ( ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ ( 𝑢 ⊼𝑔 𝑣 ) = ( 𝑎 ⊼𝑔 𝑏 ) ∧ ∀ 𝑗 ∈ ω ¬ ( 𝑢 ⊼𝑔 𝑣 ) = ∀𝑔 𝑗 𝑎 ) ) |
54 |
18 53
|
jca |
⊢ ( ( ( 𝑁 ∈ ω ∧ 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) ∧ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) ) → ( ¬ ( 𝑢 ⊼𝑔 𝑣 ) ∈ ( Fmla ‘ 𝑁 ) ∧ ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ( ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ ( 𝑢 ⊼𝑔 𝑣 ) = ( 𝑎 ⊼𝑔 𝑏 ) ∧ ∀ 𝑗 ∈ ω ¬ ( 𝑢 ⊼𝑔 𝑣 ) = ∀𝑔 𝑗 𝑎 ) ) ) |
55 |
|
eleq1 |
⊢ ( 𝑓 = ( 𝑢 ⊼𝑔 𝑣 ) → ( 𝑓 ∈ ( Fmla ‘ 𝑁 ) ↔ ( 𝑢 ⊼𝑔 𝑣 ) ∈ ( Fmla ‘ 𝑁 ) ) ) |
56 |
55
|
notbid |
⊢ ( 𝑓 = ( 𝑢 ⊼𝑔 𝑣 ) → ( ¬ 𝑓 ∈ ( Fmla ‘ 𝑁 ) ↔ ¬ ( 𝑢 ⊼𝑔 𝑣 ) ∈ ( Fmla ‘ 𝑁 ) ) ) |
57 |
|
eqeq1 |
⊢ ( 𝑓 = ( 𝑢 ⊼𝑔 𝑣 ) → ( 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ↔ ( 𝑢 ⊼𝑔 𝑣 ) = ( 𝑎 ⊼𝑔 𝑏 ) ) ) |
58 |
57
|
notbid |
⊢ ( 𝑓 = ( 𝑢 ⊼𝑔 𝑣 ) → ( ¬ 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ↔ ¬ ( 𝑢 ⊼𝑔 𝑣 ) = ( 𝑎 ⊼𝑔 𝑏 ) ) ) |
59 |
58
|
ralbidv |
⊢ ( 𝑓 = ( 𝑢 ⊼𝑔 𝑣 ) → ( ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ↔ ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ ( 𝑢 ⊼𝑔 𝑣 ) = ( 𝑎 ⊼𝑔 𝑏 ) ) ) |
60 |
|
eqeq1 |
⊢ ( 𝑓 = ( 𝑢 ⊼𝑔 𝑣 ) → ( 𝑓 = ∀𝑔 𝑗 𝑎 ↔ ( 𝑢 ⊼𝑔 𝑣 ) = ∀𝑔 𝑗 𝑎 ) ) |
61 |
60
|
notbid |
⊢ ( 𝑓 = ( 𝑢 ⊼𝑔 𝑣 ) → ( ¬ 𝑓 = ∀𝑔 𝑗 𝑎 ↔ ¬ ( 𝑢 ⊼𝑔 𝑣 ) = ∀𝑔 𝑗 𝑎 ) ) |
62 |
61
|
ralbidv |
⊢ ( 𝑓 = ( 𝑢 ⊼𝑔 𝑣 ) → ( ∀ 𝑗 ∈ ω ¬ 𝑓 = ∀𝑔 𝑗 𝑎 ↔ ∀ 𝑗 ∈ ω ¬ ( 𝑢 ⊼𝑔 𝑣 ) = ∀𝑔 𝑗 𝑎 ) ) |
63 |
59 62
|
anbi12d |
⊢ ( 𝑓 = ( 𝑢 ⊼𝑔 𝑣 ) → ( ( ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ∧ ∀ 𝑗 ∈ ω ¬ 𝑓 = ∀𝑔 𝑗 𝑎 ) ↔ ( ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ ( 𝑢 ⊼𝑔 𝑣 ) = ( 𝑎 ⊼𝑔 𝑏 ) ∧ ∀ 𝑗 ∈ ω ¬ ( 𝑢 ⊼𝑔 𝑣 ) = ∀𝑔 𝑗 𝑎 ) ) ) |
64 |
63
|
ralbidv |
⊢ ( 𝑓 = ( 𝑢 ⊼𝑔 𝑣 ) → ( ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ( ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ∧ ∀ 𝑗 ∈ ω ¬ 𝑓 = ∀𝑔 𝑗 𝑎 ) ↔ ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ( ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ ( 𝑢 ⊼𝑔 𝑣 ) = ( 𝑎 ⊼𝑔 𝑏 ) ∧ ∀ 𝑗 ∈ ω ¬ ( 𝑢 ⊼𝑔 𝑣 ) = ∀𝑔 𝑗 𝑎 ) ) ) |
65 |
56 64
|
anbi12d |
⊢ ( 𝑓 = ( 𝑢 ⊼𝑔 𝑣 ) → ( ( ¬ 𝑓 ∈ ( Fmla ‘ 𝑁 ) ∧ ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ( ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ∧ ∀ 𝑗 ∈ ω ¬ 𝑓 = ∀𝑔 𝑗 𝑎 ) ) ↔ ( ¬ ( 𝑢 ⊼𝑔 𝑣 ) ∈ ( Fmla ‘ 𝑁 ) ∧ ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ( ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ ( 𝑢 ⊼𝑔 𝑣 ) = ( 𝑎 ⊼𝑔 𝑏 ) ∧ ∀ 𝑗 ∈ ω ¬ ( 𝑢 ⊼𝑔 𝑣 ) = ∀𝑔 𝑗 𝑎 ) ) ) ) |
66 |
54 65
|
syl5ibrcom |
⊢ ( ( ( 𝑁 ∈ ω ∧ 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) ∧ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) ) → ( 𝑓 = ( 𝑢 ⊼𝑔 𝑣 ) → ( ¬ 𝑓 ∈ ( Fmla ‘ 𝑁 ) ∧ ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ( ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ∧ ∀ 𝑗 ∈ ω ¬ 𝑓 = ∀𝑔 𝑗 𝑎 ) ) ) ) |
67 |
66
|
rexlimdva |
⊢ ( ( 𝑁 ∈ ω ∧ 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) → ( ∃ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) 𝑓 = ( 𝑢 ⊼𝑔 𝑣 ) → ( ¬ 𝑓 ∈ ( Fmla ‘ 𝑁 ) ∧ ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ( ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ∧ ∀ 𝑗 ∈ ω ¬ 𝑓 = ∀𝑔 𝑗 𝑎 ) ) ) ) |
68 |
|
goalr |
⊢ ( ( 𝑁 ∈ ω ∧ ∀𝑔 𝑖 𝑢 ∈ ( Fmla ‘ 𝑁 ) ) → 𝑢 ∈ ( Fmla ‘ 𝑁 ) ) |
69 |
68 12
|
syl |
⊢ ( ( 𝑁 ∈ ω ∧ ∀𝑔 𝑖 𝑢 ∈ ( Fmla ‘ 𝑁 ) ) → ¬ 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) |
70 |
69
|
ex |
⊢ ( 𝑁 ∈ ω → ( ∀𝑔 𝑖 𝑢 ∈ ( Fmla ‘ 𝑁 ) → ¬ 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) ) |
71 |
70
|
con2d |
⊢ ( 𝑁 ∈ ω → ( 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) → ¬ ∀𝑔 𝑖 𝑢 ∈ ( Fmla ‘ 𝑁 ) ) ) |
72 |
71
|
imp |
⊢ ( ( 𝑁 ∈ ω ∧ 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) → ¬ ∀𝑔 𝑖 𝑢 ∈ ( Fmla ‘ 𝑁 ) ) |
73 |
72
|
adantr |
⊢ ( ( ( 𝑁 ∈ ω ∧ 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) ∧ 𝑖 ∈ ω ) → ¬ ∀𝑔 𝑖 𝑢 ∈ ( Fmla ‘ 𝑁 ) ) |
74 |
|
gonanegoal |
⊢ ( 𝑎 ⊼𝑔 𝑏 ) ≠ ∀𝑔 𝑖 𝑢 |
75 |
74
|
nesymi |
⊢ ¬ ∀𝑔 𝑖 𝑢 = ( 𝑎 ⊼𝑔 𝑏 ) |
76 |
75
|
a1i |
⊢ ( ( ( 𝑁 ∈ ω ∧ 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) ∧ 𝑖 ∈ ω ) → ¬ ∀𝑔 𝑖 𝑢 = ( 𝑎 ⊼𝑔 𝑏 ) ) |
77 |
76
|
ralrimivw |
⊢ ( ( ( 𝑁 ∈ ω ∧ 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) ∧ 𝑖 ∈ ω ) → ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ ∀𝑔 𝑖 𝑢 = ( 𝑎 ⊼𝑔 𝑏 ) ) |
78 |
77
|
ralrimivw |
⊢ ( ( ( 𝑁 ∈ ω ∧ 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) ∧ 𝑖 ∈ ω ) → ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ ∀𝑔 𝑖 𝑢 = ( 𝑎 ⊼𝑔 𝑏 ) ) |
79 |
22
|
olcd |
⊢ ( ( 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ∧ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ) → ( ¬ 𝑖 = 𝑗 ∨ ¬ 𝑢 = 𝑎 ) ) |
80 |
|
ianor |
⊢ ( ¬ ( 𝑖 = 𝑗 ∧ 𝑢 = 𝑎 ) ↔ ( ¬ 𝑖 = 𝑗 ∨ ¬ 𝑢 = 𝑎 ) ) |
81 |
|
vex |
⊢ 𝑖 ∈ V |
82 |
81 25
|
opth |
⊢ ( 〈 𝑖 , 𝑢 〉 = 〈 𝑗 , 𝑎 〉 ↔ ( 𝑖 = 𝑗 ∧ 𝑢 = 𝑎 ) ) |
83 |
80 82
|
xchnxbir |
⊢ ( ¬ 〈 𝑖 , 𝑢 〉 = 〈 𝑗 , 𝑎 〉 ↔ ( ¬ 𝑖 = 𝑗 ∨ ¬ 𝑢 = 𝑎 ) ) |
84 |
79 83
|
sylibr |
⊢ ( ( 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ∧ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ) → ¬ 〈 𝑖 , 𝑢 〉 = 〈 𝑗 , 𝑎 〉 ) |
85 |
84
|
olcd |
⊢ ( ( 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ∧ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ) → ( ¬ 2o = 2o ∨ ¬ 〈 𝑖 , 𝑢 〉 = 〈 𝑗 , 𝑎 〉 ) ) |
86 |
|
ianor |
⊢ ( ¬ ( 2o = 2o ∧ 〈 𝑖 , 𝑢 〉 = 〈 𝑗 , 𝑎 〉 ) ↔ ( ¬ 2o = 2o ∨ ¬ 〈 𝑖 , 𝑢 〉 = 〈 𝑗 , 𝑎 〉 ) ) |
87 |
|
2oex |
⊢ 2o ∈ V |
88 |
|
opex |
⊢ 〈 𝑖 , 𝑢 〉 ∈ V |
89 |
87 88
|
opth |
⊢ ( 〈 2o , 〈 𝑖 , 𝑢 〉 〉 = 〈 2o , 〈 𝑗 , 𝑎 〉 〉 ↔ ( 2o = 2o ∧ 〈 𝑖 , 𝑢 〉 = 〈 𝑗 , 𝑎 〉 ) ) |
90 |
86 89
|
xchnxbir |
⊢ ( ¬ 〈 2o , 〈 𝑖 , 𝑢 〉 〉 = 〈 2o , 〈 𝑗 , 𝑎 〉 〉 ↔ ( ¬ 2o = 2o ∨ ¬ 〈 𝑖 , 𝑢 〉 = 〈 𝑗 , 𝑎 〉 ) ) |
91 |
|
df-goal |
⊢ ∀𝑔 𝑖 𝑢 = 〈 2o , 〈 𝑖 , 𝑢 〉 〉 |
92 |
|
df-goal |
⊢ ∀𝑔 𝑗 𝑎 = 〈 2o , 〈 𝑗 , 𝑎 〉 〉 |
93 |
91 92
|
eqeq12i |
⊢ ( ∀𝑔 𝑖 𝑢 = ∀𝑔 𝑗 𝑎 ↔ 〈 2o , 〈 𝑖 , 𝑢 〉 〉 = 〈 2o , 〈 𝑗 , 𝑎 〉 〉 ) |
94 |
90 93
|
xchnxbir |
⊢ ( ¬ ∀𝑔 𝑖 𝑢 = ∀𝑔 𝑗 𝑎 ↔ ( ¬ 2o = 2o ∨ ¬ 〈 𝑖 , 𝑢 〉 = 〈 𝑗 , 𝑎 〉 ) ) |
95 |
85 94
|
sylibr |
⊢ ( ( 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ∧ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ) → ¬ ∀𝑔 𝑖 𝑢 = ∀𝑔 𝑗 𝑎 ) |
96 |
95
|
ralrimivw |
⊢ ( ( 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ∧ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ) → ∀ 𝑗 ∈ ω ¬ ∀𝑔 𝑖 𝑢 = ∀𝑔 𝑗 𝑎 ) |
97 |
96
|
ralrimiva |
⊢ ( 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) → ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ∀ 𝑗 ∈ ω ¬ ∀𝑔 𝑖 𝑢 = ∀𝑔 𝑗 𝑎 ) |
98 |
97
|
adantl |
⊢ ( ( 𝑁 ∈ ω ∧ 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) → ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ∀ 𝑗 ∈ ω ¬ ∀𝑔 𝑖 𝑢 = ∀𝑔 𝑗 𝑎 ) |
99 |
98
|
adantr |
⊢ ( ( ( 𝑁 ∈ ω ∧ 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) ∧ 𝑖 ∈ ω ) → ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ∀ 𝑗 ∈ ω ¬ ∀𝑔 𝑖 𝑢 = ∀𝑔 𝑗 𝑎 ) |
100 |
|
r19.26 |
⊢ ( ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ( ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ ∀𝑔 𝑖 𝑢 = ( 𝑎 ⊼𝑔 𝑏 ) ∧ ∀ 𝑗 ∈ ω ¬ ∀𝑔 𝑖 𝑢 = ∀𝑔 𝑗 𝑎 ) ↔ ( ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ ∀𝑔 𝑖 𝑢 = ( 𝑎 ⊼𝑔 𝑏 ) ∧ ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ∀ 𝑗 ∈ ω ¬ ∀𝑔 𝑖 𝑢 = ∀𝑔 𝑗 𝑎 ) ) |
101 |
78 99 100
|
sylanbrc |
⊢ ( ( ( 𝑁 ∈ ω ∧ 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) ∧ 𝑖 ∈ ω ) → ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ( ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ ∀𝑔 𝑖 𝑢 = ( 𝑎 ⊼𝑔 𝑏 ) ∧ ∀ 𝑗 ∈ ω ¬ ∀𝑔 𝑖 𝑢 = ∀𝑔 𝑗 𝑎 ) ) |
102 |
73 101
|
jca |
⊢ ( ( ( 𝑁 ∈ ω ∧ 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) ∧ 𝑖 ∈ ω ) → ( ¬ ∀𝑔 𝑖 𝑢 ∈ ( Fmla ‘ 𝑁 ) ∧ ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ( ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ ∀𝑔 𝑖 𝑢 = ( 𝑎 ⊼𝑔 𝑏 ) ∧ ∀ 𝑗 ∈ ω ¬ ∀𝑔 𝑖 𝑢 = ∀𝑔 𝑗 𝑎 ) ) ) |
103 |
|
eleq1 |
⊢ ( ∀𝑔 𝑖 𝑢 = 𝑓 → ( ∀𝑔 𝑖 𝑢 ∈ ( Fmla ‘ 𝑁 ) ↔ 𝑓 ∈ ( Fmla ‘ 𝑁 ) ) ) |
104 |
103
|
notbid |
⊢ ( ∀𝑔 𝑖 𝑢 = 𝑓 → ( ¬ ∀𝑔 𝑖 𝑢 ∈ ( Fmla ‘ 𝑁 ) ↔ ¬ 𝑓 ∈ ( Fmla ‘ 𝑁 ) ) ) |
105 |
|
eqeq1 |
⊢ ( ∀𝑔 𝑖 𝑢 = 𝑓 → ( ∀𝑔 𝑖 𝑢 = ( 𝑎 ⊼𝑔 𝑏 ) ↔ 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ) ) |
106 |
105
|
notbid |
⊢ ( ∀𝑔 𝑖 𝑢 = 𝑓 → ( ¬ ∀𝑔 𝑖 𝑢 = ( 𝑎 ⊼𝑔 𝑏 ) ↔ ¬ 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ) ) |
107 |
106
|
ralbidv |
⊢ ( ∀𝑔 𝑖 𝑢 = 𝑓 → ( ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ ∀𝑔 𝑖 𝑢 = ( 𝑎 ⊼𝑔 𝑏 ) ↔ ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ) ) |
108 |
|
eqeq1 |
⊢ ( ∀𝑔 𝑖 𝑢 = 𝑓 → ( ∀𝑔 𝑖 𝑢 = ∀𝑔 𝑗 𝑎 ↔ 𝑓 = ∀𝑔 𝑗 𝑎 ) ) |
109 |
108
|
notbid |
⊢ ( ∀𝑔 𝑖 𝑢 = 𝑓 → ( ¬ ∀𝑔 𝑖 𝑢 = ∀𝑔 𝑗 𝑎 ↔ ¬ 𝑓 = ∀𝑔 𝑗 𝑎 ) ) |
110 |
109
|
ralbidv |
⊢ ( ∀𝑔 𝑖 𝑢 = 𝑓 → ( ∀ 𝑗 ∈ ω ¬ ∀𝑔 𝑖 𝑢 = ∀𝑔 𝑗 𝑎 ↔ ∀ 𝑗 ∈ ω ¬ 𝑓 = ∀𝑔 𝑗 𝑎 ) ) |
111 |
107 110
|
anbi12d |
⊢ ( ∀𝑔 𝑖 𝑢 = 𝑓 → ( ( ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ ∀𝑔 𝑖 𝑢 = ( 𝑎 ⊼𝑔 𝑏 ) ∧ ∀ 𝑗 ∈ ω ¬ ∀𝑔 𝑖 𝑢 = ∀𝑔 𝑗 𝑎 ) ↔ ( ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ∧ ∀ 𝑗 ∈ ω ¬ 𝑓 = ∀𝑔 𝑗 𝑎 ) ) ) |
112 |
111
|
ralbidv |
⊢ ( ∀𝑔 𝑖 𝑢 = 𝑓 → ( ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ( ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ ∀𝑔 𝑖 𝑢 = ( 𝑎 ⊼𝑔 𝑏 ) ∧ ∀ 𝑗 ∈ ω ¬ ∀𝑔 𝑖 𝑢 = ∀𝑔 𝑗 𝑎 ) ↔ ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ( ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ∧ ∀ 𝑗 ∈ ω ¬ 𝑓 = ∀𝑔 𝑗 𝑎 ) ) ) |
113 |
104 112
|
anbi12d |
⊢ ( ∀𝑔 𝑖 𝑢 = 𝑓 → ( ( ¬ ∀𝑔 𝑖 𝑢 ∈ ( Fmla ‘ 𝑁 ) ∧ ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ( ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ ∀𝑔 𝑖 𝑢 = ( 𝑎 ⊼𝑔 𝑏 ) ∧ ∀ 𝑗 ∈ ω ¬ ∀𝑔 𝑖 𝑢 = ∀𝑔 𝑗 𝑎 ) ) ↔ ( ¬ 𝑓 ∈ ( Fmla ‘ 𝑁 ) ∧ ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ( ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ∧ ∀ 𝑗 ∈ ω ¬ 𝑓 = ∀𝑔 𝑗 𝑎 ) ) ) ) |
114 |
113
|
eqcoms |
⊢ ( 𝑓 = ∀𝑔 𝑖 𝑢 → ( ( ¬ ∀𝑔 𝑖 𝑢 ∈ ( Fmla ‘ 𝑁 ) ∧ ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ( ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ ∀𝑔 𝑖 𝑢 = ( 𝑎 ⊼𝑔 𝑏 ) ∧ ∀ 𝑗 ∈ ω ¬ ∀𝑔 𝑖 𝑢 = ∀𝑔 𝑗 𝑎 ) ) ↔ ( ¬ 𝑓 ∈ ( Fmla ‘ 𝑁 ) ∧ ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ( ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ∧ ∀ 𝑗 ∈ ω ¬ 𝑓 = ∀𝑔 𝑗 𝑎 ) ) ) ) |
115 |
102 114
|
syl5ibcom |
⊢ ( ( ( 𝑁 ∈ ω ∧ 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) ∧ 𝑖 ∈ ω ) → ( 𝑓 = ∀𝑔 𝑖 𝑢 → ( ¬ 𝑓 ∈ ( Fmla ‘ 𝑁 ) ∧ ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ( ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ∧ ∀ 𝑗 ∈ ω ¬ 𝑓 = ∀𝑔 𝑗 𝑎 ) ) ) ) |
116 |
115
|
rexlimdva |
⊢ ( ( 𝑁 ∈ ω ∧ 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) → ( ∃ 𝑖 ∈ ω 𝑓 = ∀𝑔 𝑖 𝑢 → ( ¬ 𝑓 ∈ ( Fmla ‘ 𝑁 ) ∧ ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ( ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ∧ ∀ 𝑗 ∈ ω ¬ 𝑓 = ∀𝑔 𝑗 𝑎 ) ) ) ) |
117 |
67 116
|
jaod |
⊢ ( ( 𝑁 ∈ ω ∧ 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) → ( ( ∃ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) 𝑓 = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω 𝑓 = ∀𝑔 𝑖 𝑢 ) → ( ¬ 𝑓 ∈ ( Fmla ‘ 𝑁 ) ∧ ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ( ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ∧ ∀ 𝑗 ∈ ω ¬ 𝑓 = ∀𝑔 𝑗 𝑎 ) ) ) ) |
118 |
117
|
rexlimdva |
⊢ ( 𝑁 ∈ ω → ( ∃ 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) 𝑓 = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω 𝑓 = ∀𝑔 𝑖 𝑢 ) → ( ¬ 𝑓 ∈ ( Fmla ‘ 𝑁 ) ∧ ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ( ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ∧ ∀ 𝑗 ∈ ω ¬ 𝑓 = ∀𝑔 𝑗 𝑎 ) ) ) ) |
119 |
|
elndif |
⊢ ( 𝑣 ∈ ( Fmla ‘ 𝑁 ) → ¬ 𝑣 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) |
120 |
119
|
adantl |
⊢ ( ( 𝑢 ∈ ( Fmla ‘ 𝑁 ) ∧ 𝑣 ∈ ( Fmla ‘ 𝑁 ) ) → ¬ 𝑣 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) |
121 |
120
|
intnand |
⊢ ( ( 𝑢 ∈ ( Fmla ‘ 𝑁 ) ∧ 𝑣 ∈ ( Fmla ‘ 𝑁 ) ) → ¬ ( 𝑢 ∈ ( Fmla ‘ 𝑁 ) ∧ 𝑣 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) ) |
122 |
11 121
|
syl |
⊢ ( ( 𝑁 ∈ ω ∧ ( 𝑢 ⊼𝑔 𝑣 ) ∈ ( Fmla ‘ 𝑁 ) ) → ¬ ( 𝑢 ∈ ( Fmla ‘ 𝑁 ) ∧ 𝑣 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) ) |
123 |
122
|
ex |
⊢ ( 𝑁 ∈ ω → ( ( 𝑢 ⊼𝑔 𝑣 ) ∈ ( Fmla ‘ 𝑁 ) → ¬ ( 𝑢 ∈ ( Fmla ‘ 𝑁 ) ∧ 𝑣 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) ) ) |
124 |
123
|
con2d |
⊢ ( 𝑁 ∈ ω → ( ( 𝑢 ∈ ( Fmla ‘ 𝑁 ) ∧ 𝑣 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) → ¬ ( 𝑢 ⊼𝑔 𝑣 ) ∈ ( Fmla ‘ 𝑁 ) ) ) |
125 |
124
|
impl |
⊢ ( ( ( 𝑁 ∈ ω ∧ 𝑢 ∈ ( Fmla ‘ 𝑁 ) ) ∧ 𝑣 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) → ¬ ( 𝑢 ⊼𝑔 𝑣 ) ∈ ( Fmla ‘ 𝑁 ) ) |
126 |
|
elneeldif |
⊢ ( ( 𝑏 ∈ ( Fmla ‘ 𝑁 ) ∧ 𝑣 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) → 𝑏 ≠ 𝑣 ) |
127 |
126
|
necomd |
⊢ ( ( 𝑏 ∈ ( Fmla ‘ 𝑁 ) ∧ 𝑣 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) → 𝑣 ≠ 𝑏 ) |
128 |
127
|
ancoms |
⊢ ( ( 𝑣 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ∧ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ) → 𝑣 ≠ 𝑏 ) |
129 |
128
|
neneqd |
⊢ ( ( 𝑣 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ∧ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ) → ¬ 𝑣 = 𝑏 ) |
130 |
129
|
olcd |
⊢ ( ( 𝑣 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ∧ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ) → ( ¬ 𝑢 = 𝑎 ∨ ¬ 𝑣 = 𝑏 ) ) |
131 |
130 28
|
sylibr |
⊢ ( ( 𝑣 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ∧ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ) → ¬ 〈 𝑢 , 𝑣 〉 = 〈 𝑎 , 𝑏 〉 ) |
132 |
131
|
intnand |
⊢ ( ( 𝑣 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ∧ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ) → ¬ ( 1o = 1o ∧ 〈 𝑢 , 𝑣 〉 = 〈 𝑎 , 𝑏 〉 ) ) |
133 |
132 40
|
sylnibr |
⊢ ( ( 𝑣 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ∧ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ) → ¬ ( 𝑢 ⊼𝑔 𝑣 ) = ( 𝑎 ⊼𝑔 𝑏 ) ) |
134 |
133
|
ralrimiva |
⊢ ( 𝑣 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) → ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ ( 𝑢 ⊼𝑔 𝑣 ) = ( 𝑎 ⊼𝑔 𝑏 ) ) |
135 |
134
|
ralrimivw |
⊢ ( 𝑣 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) → ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ ( 𝑢 ⊼𝑔 𝑣 ) = ( 𝑎 ⊼𝑔 𝑏 ) ) |
136 |
135
|
adantl |
⊢ ( ( ( 𝑁 ∈ ω ∧ 𝑢 ∈ ( Fmla ‘ 𝑁 ) ) ∧ 𝑣 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) → ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ ( 𝑢 ⊼𝑔 𝑣 ) = ( 𝑎 ⊼𝑔 𝑏 ) ) |
137 |
48
|
a1i |
⊢ ( ( ( 𝑁 ∈ ω ∧ 𝑢 ∈ ( Fmla ‘ 𝑁 ) ) ∧ 𝑣 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) → ¬ ( 𝑢 ⊼𝑔 𝑣 ) = ∀𝑔 𝑗 𝑎 ) |
138 |
137
|
ralrimivw |
⊢ ( ( ( 𝑁 ∈ ω ∧ 𝑢 ∈ ( Fmla ‘ 𝑁 ) ) ∧ 𝑣 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) → ∀ 𝑗 ∈ ω ¬ ( 𝑢 ⊼𝑔 𝑣 ) = ∀𝑔 𝑗 𝑎 ) |
139 |
138
|
ralrimivw |
⊢ ( ( ( 𝑁 ∈ ω ∧ 𝑢 ∈ ( Fmla ‘ 𝑁 ) ) ∧ 𝑣 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) → ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ∀ 𝑗 ∈ ω ¬ ( 𝑢 ⊼𝑔 𝑣 ) = ∀𝑔 𝑗 𝑎 ) |
140 |
136 139 52
|
sylanbrc |
⊢ ( ( ( 𝑁 ∈ ω ∧ 𝑢 ∈ ( Fmla ‘ 𝑁 ) ) ∧ 𝑣 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) → ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ( ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ ( 𝑢 ⊼𝑔 𝑣 ) = ( 𝑎 ⊼𝑔 𝑏 ) ∧ ∀ 𝑗 ∈ ω ¬ ( 𝑢 ⊼𝑔 𝑣 ) = ∀𝑔 𝑗 𝑎 ) ) |
141 |
125 140
|
jca |
⊢ ( ( ( 𝑁 ∈ ω ∧ 𝑢 ∈ ( Fmla ‘ 𝑁 ) ) ∧ 𝑣 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) → ( ¬ ( 𝑢 ⊼𝑔 𝑣 ) ∈ ( Fmla ‘ 𝑁 ) ∧ ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ( ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ ( 𝑢 ⊼𝑔 𝑣 ) = ( 𝑎 ⊼𝑔 𝑏 ) ∧ ∀ 𝑗 ∈ ω ¬ ( 𝑢 ⊼𝑔 𝑣 ) = ∀𝑔 𝑗 𝑎 ) ) ) |
142 |
|
eleq1 |
⊢ ( ( 𝑢 ⊼𝑔 𝑣 ) = 𝑓 → ( ( 𝑢 ⊼𝑔 𝑣 ) ∈ ( Fmla ‘ 𝑁 ) ↔ 𝑓 ∈ ( Fmla ‘ 𝑁 ) ) ) |
143 |
142
|
notbid |
⊢ ( ( 𝑢 ⊼𝑔 𝑣 ) = 𝑓 → ( ¬ ( 𝑢 ⊼𝑔 𝑣 ) ∈ ( Fmla ‘ 𝑁 ) ↔ ¬ 𝑓 ∈ ( Fmla ‘ 𝑁 ) ) ) |
144 |
|
eqeq1 |
⊢ ( ( 𝑢 ⊼𝑔 𝑣 ) = 𝑓 → ( ( 𝑢 ⊼𝑔 𝑣 ) = ( 𝑎 ⊼𝑔 𝑏 ) ↔ 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ) ) |
145 |
144
|
notbid |
⊢ ( ( 𝑢 ⊼𝑔 𝑣 ) = 𝑓 → ( ¬ ( 𝑢 ⊼𝑔 𝑣 ) = ( 𝑎 ⊼𝑔 𝑏 ) ↔ ¬ 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ) ) |
146 |
145
|
ralbidv |
⊢ ( ( 𝑢 ⊼𝑔 𝑣 ) = 𝑓 → ( ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ ( 𝑢 ⊼𝑔 𝑣 ) = ( 𝑎 ⊼𝑔 𝑏 ) ↔ ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ) ) |
147 |
|
eqeq1 |
⊢ ( ( 𝑢 ⊼𝑔 𝑣 ) = 𝑓 → ( ( 𝑢 ⊼𝑔 𝑣 ) = ∀𝑔 𝑗 𝑎 ↔ 𝑓 = ∀𝑔 𝑗 𝑎 ) ) |
148 |
147
|
notbid |
⊢ ( ( 𝑢 ⊼𝑔 𝑣 ) = 𝑓 → ( ¬ ( 𝑢 ⊼𝑔 𝑣 ) = ∀𝑔 𝑗 𝑎 ↔ ¬ 𝑓 = ∀𝑔 𝑗 𝑎 ) ) |
149 |
148
|
ralbidv |
⊢ ( ( 𝑢 ⊼𝑔 𝑣 ) = 𝑓 → ( ∀ 𝑗 ∈ ω ¬ ( 𝑢 ⊼𝑔 𝑣 ) = ∀𝑔 𝑗 𝑎 ↔ ∀ 𝑗 ∈ ω ¬ 𝑓 = ∀𝑔 𝑗 𝑎 ) ) |
150 |
146 149
|
anbi12d |
⊢ ( ( 𝑢 ⊼𝑔 𝑣 ) = 𝑓 → ( ( ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ ( 𝑢 ⊼𝑔 𝑣 ) = ( 𝑎 ⊼𝑔 𝑏 ) ∧ ∀ 𝑗 ∈ ω ¬ ( 𝑢 ⊼𝑔 𝑣 ) = ∀𝑔 𝑗 𝑎 ) ↔ ( ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ∧ ∀ 𝑗 ∈ ω ¬ 𝑓 = ∀𝑔 𝑗 𝑎 ) ) ) |
151 |
150
|
ralbidv |
⊢ ( ( 𝑢 ⊼𝑔 𝑣 ) = 𝑓 → ( ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ( ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ ( 𝑢 ⊼𝑔 𝑣 ) = ( 𝑎 ⊼𝑔 𝑏 ) ∧ ∀ 𝑗 ∈ ω ¬ ( 𝑢 ⊼𝑔 𝑣 ) = ∀𝑔 𝑗 𝑎 ) ↔ ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ( ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ∧ ∀ 𝑗 ∈ ω ¬ 𝑓 = ∀𝑔 𝑗 𝑎 ) ) ) |
152 |
143 151
|
anbi12d |
⊢ ( ( 𝑢 ⊼𝑔 𝑣 ) = 𝑓 → ( ( ¬ ( 𝑢 ⊼𝑔 𝑣 ) ∈ ( Fmla ‘ 𝑁 ) ∧ ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ( ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ ( 𝑢 ⊼𝑔 𝑣 ) = ( 𝑎 ⊼𝑔 𝑏 ) ∧ ∀ 𝑗 ∈ ω ¬ ( 𝑢 ⊼𝑔 𝑣 ) = ∀𝑔 𝑗 𝑎 ) ) ↔ ( ¬ 𝑓 ∈ ( Fmla ‘ 𝑁 ) ∧ ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ( ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ∧ ∀ 𝑗 ∈ ω ¬ 𝑓 = ∀𝑔 𝑗 𝑎 ) ) ) ) |
153 |
152
|
eqcoms |
⊢ ( 𝑓 = ( 𝑢 ⊼𝑔 𝑣 ) → ( ( ¬ ( 𝑢 ⊼𝑔 𝑣 ) ∈ ( Fmla ‘ 𝑁 ) ∧ ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ( ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ ( 𝑢 ⊼𝑔 𝑣 ) = ( 𝑎 ⊼𝑔 𝑏 ) ∧ ∀ 𝑗 ∈ ω ¬ ( 𝑢 ⊼𝑔 𝑣 ) = ∀𝑔 𝑗 𝑎 ) ) ↔ ( ¬ 𝑓 ∈ ( Fmla ‘ 𝑁 ) ∧ ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ( ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ∧ ∀ 𝑗 ∈ ω ¬ 𝑓 = ∀𝑔 𝑗 𝑎 ) ) ) ) |
154 |
141 153
|
syl5ibcom |
⊢ ( ( ( 𝑁 ∈ ω ∧ 𝑢 ∈ ( Fmla ‘ 𝑁 ) ) ∧ 𝑣 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) → ( 𝑓 = ( 𝑢 ⊼𝑔 𝑣 ) → ( ¬ 𝑓 ∈ ( Fmla ‘ 𝑁 ) ∧ ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ( ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ∧ ∀ 𝑗 ∈ ω ¬ 𝑓 = ∀𝑔 𝑗 𝑎 ) ) ) ) |
155 |
154
|
rexlimdva |
⊢ ( ( 𝑁 ∈ ω ∧ 𝑢 ∈ ( Fmla ‘ 𝑁 ) ) → ( ∃ 𝑣 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑓 = ( 𝑢 ⊼𝑔 𝑣 ) → ( ¬ 𝑓 ∈ ( Fmla ‘ 𝑁 ) ∧ ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ( ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ∧ ∀ 𝑗 ∈ ω ¬ 𝑓 = ∀𝑔 𝑗 𝑎 ) ) ) ) |
156 |
155
|
rexlimdva |
⊢ ( 𝑁 ∈ ω → ( ∃ 𝑢 ∈ ( Fmla ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑓 = ( 𝑢 ⊼𝑔 𝑣 ) → ( ¬ 𝑓 ∈ ( Fmla ‘ 𝑁 ) ∧ ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ( ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ∧ ∀ 𝑗 ∈ ω ¬ 𝑓 = ∀𝑔 𝑗 𝑎 ) ) ) ) |
157 |
118 156
|
jaod |
⊢ ( 𝑁 ∈ ω → ( ( ∃ 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) 𝑓 = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω 𝑓 = ∀𝑔 𝑖 𝑢 ) ∨ ∃ 𝑢 ∈ ( Fmla ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑓 = ( 𝑢 ⊼𝑔 𝑣 ) ) → ( ¬ 𝑓 ∈ ( Fmla ‘ 𝑁 ) ∧ ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ( ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ∧ ∀ 𝑗 ∈ ω ¬ 𝑓 = ∀𝑔 𝑗 𝑎 ) ) ) ) |
158 |
|
isfmlasuc |
⊢ ( ( 𝑁 ∈ ω ∧ 𝑓 ∈ V ) → ( 𝑓 ∈ ( Fmla ‘ suc 𝑁 ) ↔ ( 𝑓 ∈ ( Fmla ‘ 𝑁 ) ∨ ∃ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ( ∃ 𝑏 ∈ ( Fmla ‘ 𝑁 ) 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ∨ ∃ 𝑗 ∈ ω 𝑓 = ∀𝑔 𝑗 𝑎 ) ) ) ) |
159 |
158
|
elvd |
⊢ ( 𝑁 ∈ ω → ( 𝑓 ∈ ( Fmla ‘ suc 𝑁 ) ↔ ( 𝑓 ∈ ( Fmla ‘ 𝑁 ) ∨ ∃ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ( ∃ 𝑏 ∈ ( Fmla ‘ 𝑁 ) 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ∨ ∃ 𝑗 ∈ ω 𝑓 = ∀𝑔 𝑗 𝑎 ) ) ) ) |
160 |
159
|
notbid |
⊢ ( 𝑁 ∈ ω → ( ¬ 𝑓 ∈ ( Fmla ‘ suc 𝑁 ) ↔ ¬ ( 𝑓 ∈ ( Fmla ‘ 𝑁 ) ∨ ∃ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ( ∃ 𝑏 ∈ ( Fmla ‘ 𝑁 ) 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ∨ ∃ 𝑗 ∈ ω 𝑓 = ∀𝑔 𝑗 𝑎 ) ) ) ) |
161 |
|
ioran |
⊢ ( ¬ ( 𝑓 ∈ ( Fmla ‘ 𝑁 ) ∨ ∃ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ( ∃ 𝑏 ∈ ( Fmla ‘ 𝑁 ) 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ∨ ∃ 𝑗 ∈ ω 𝑓 = ∀𝑔 𝑗 𝑎 ) ) ↔ ( ¬ 𝑓 ∈ ( Fmla ‘ 𝑁 ) ∧ ¬ ∃ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ( ∃ 𝑏 ∈ ( Fmla ‘ 𝑁 ) 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ∨ ∃ 𝑗 ∈ ω 𝑓 = ∀𝑔 𝑗 𝑎 ) ) ) |
162 |
|
ralnex |
⊢ ( ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ↔ ¬ ∃ 𝑏 ∈ ( Fmla ‘ 𝑁 ) 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ) |
163 |
|
ralnex |
⊢ ( ∀ 𝑗 ∈ ω ¬ 𝑓 = ∀𝑔 𝑗 𝑎 ↔ ¬ ∃ 𝑗 ∈ ω 𝑓 = ∀𝑔 𝑗 𝑎 ) |
164 |
162 163
|
anbi12i |
⊢ ( ( ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ∧ ∀ 𝑗 ∈ ω ¬ 𝑓 = ∀𝑔 𝑗 𝑎 ) ↔ ( ¬ ∃ 𝑏 ∈ ( Fmla ‘ 𝑁 ) 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ∧ ¬ ∃ 𝑗 ∈ ω 𝑓 = ∀𝑔 𝑗 𝑎 ) ) |
165 |
|
ioran |
⊢ ( ¬ ( ∃ 𝑏 ∈ ( Fmla ‘ 𝑁 ) 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ∨ ∃ 𝑗 ∈ ω 𝑓 = ∀𝑔 𝑗 𝑎 ) ↔ ( ¬ ∃ 𝑏 ∈ ( Fmla ‘ 𝑁 ) 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ∧ ¬ ∃ 𝑗 ∈ ω 𝑓 = ∀𝑔 𝑗 𝑎 ) ) |
166 |
164 165
|
bitr4i |
⊢ ( ( ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ∧ ∀ 𝑗 ∈ ω ¬ 𝑓 = ∀𝑔 𝑗 𝑎 ) ↔ ¬ ( ∃ 𝑏 ∈ ( Fmla ‘ 𝑁 ) 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ∨ ∃ 𝑗 ∈ ω 𝑓 = ∀𝑔 𝑗 𝑎 ) ) |
167 |
166
|
ralbii |
⊢ ( ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ( ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ∧ ∀ 𝑗 ∈ ω ¬ 𝑓 = ∀𝑔 𝑗 𝑎 ) ↔ ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ¬ ( ∃ 𝑏 ∈ ( Fmla ‘ 𝑁 ) 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ∨ ∃ 𝑗 ∈ ω 𝑓 = ∀𝑔 𝑗 𝑎 ) ) |
168 |
|
ralnex |
⊢ ( ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ¬ ( ∃ 𝑏 ∈ ( Fmla ‘ 𝑁 ) 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ∨ ∃ 𝑗 ∈ ω 𝑓 = ∀𝑔 𝑗 𝑎 ) ↔ ¬ ∃ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ( ∃ 𝑏 ∈ ( Fmla ‘ 𝑁 ) 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ∨ ∃ 𝑗 ∈ ω 𝑓 = ∀𝑔 𝑗 𝑎 ) ) |
169 |
167 168
|
bitr2i |
⊢ ( ¬ ∃ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ( ∃ 𝑏 ∈ ( Fmla ‘ 𝑁 ) 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ∨ ∃ 𝑗 ∈ ω 𝑓 = ∀𝑔 𝑗 𝑎 ) ↔ ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ( ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ∧ ∀ 𝑗 ∈ ω ¬ 𝑓 = ∀𝑔 𝑗 𝑎 ) ) |
170 |
169
|
anbi2i |
⊢ ( ( ¬ 𝑓 ∈ ( Fmla ‘ 𝑁 ) ∧ ¬ ∃ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ( ∃ 𝑏 ∈ ( Fmla ‘ 𝑁 ) 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ∨ ∃ 𝑗 ∈ ω 𝑓 = ∀𝑔 𝑗 𝑎 ) ) ↔ ( ¬ 𝑓 ∈ ( Fmla ‘ 𝑁 ) ∧ ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ( ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ∧ ∀ 𝑗 ∈ ω ¬ 𝑓 = ∀𝑔 𝑗 𝑎 ) ) ) |
171 |
161 170
|
bitri |
⊢ ( ¬ ( 𝑓 ∈ ( Fmla ‘ 𝑁 ) ∨ ∃ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ( ∃ 𝑏 ∈ ( Fmla ‘ 𝑁 ) 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ∨ ∃ 𝑗 ∈ ω 𝑓 = ∀𝑔 𝑗 𝑎 ) ) ↔ ( ¬ 𝑓 ∈ ( Fmla ‘ 𝑁 ) ∧ ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ( ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ∧ ∀ 𝑗 ∈ ω ¬ 𝑓 = ∀𝑔 𝑗 𝑎 ) ) ) |
172 |
160 171
|
syl6bb |
⊢ ( 𝑁 ∈ ω → ( ¬ 𝑓 ∈ ( Fmla ‘ suc 𝑁 ) ↔ ( ¬ 𝑓 ∈ ( Fmla ‘ 𝑁 ) ∧ ∀ 𝑎 ∈ ( Fmla ‘ 𝑁 ) ( ∀ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ¬ 𝑓 = ( 𝑎 ⊼𝑔 𝑏 ) ∧ ∀ 𝑗 ∈ ω ¬ 𝑓 = ∀𝑔 𝑗 𝑎 ) ) ) ) |
173 |
157 172
|
sylibrd |
⊢ ( 𝑁 ∈ ω → ( ( ∃ 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) 𝑓 = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω 𝑓 = ∀𝑔 𝑖 𝑢 ) ∨ ∃ 𝑢 ∈ ( Fmla ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑓 = ( 𝑢 ⊼𝑔 𝑣 ) ) → ¬ 𝑓 ∈ ( Fmla ‘ suc 𝑁 ) ) ) |
174 |
10 173
|
syl5bi |
⊢ ( 𝑁 ∈ ω → ( 𝑓 ∈ { 𝑥 ∣ ( ∃ 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑢 ) ∨ ∃ 𝑢 ∈ ( Fmla ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑢 ⊼𝑔 𝑣 ) ) } → ¬ 𝑓 ∈ ( Fmla ‘ suc 𝑁 ) ) ) |
175 |
174
|
ralrimiv |
⊢ ( 𝑁 ∈ ω → ∀ 𝑓 ∈ { 𝑥 ∣ ( ∃ 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑢 ) ∨ ∃ 𝑢 ∈ ( Fmla ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑢 ⊼𝑔 𝑣 ) ) } ¬ 𝑓 ∈ ( Fmla ‘ suc 𝑁 ) ) |
176 |
|
disjr |
⊢ ( ( ( Fmla ‘ suc 𝑁 ) ∩ { 𝑥 ∣ ( ∃ 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑢 ) ∨ ∃ 𝑢 ∈ ( Fmla ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑢 ⊼𝑔 𝑣 ) ) } ) = ∅ ↔ ∀ 𝑓 ∈ { 𝑥 ∣ ( ∃ 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑢 ) ∨ ∃ 𝑢 ∈ ( Fmla ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑢 ⊼𝑔 𝑣 ) ) } ¬ 𝑓 ∈ ( Fmla ‘ suc 𝑁 ) ) |
177 |
175 176
|
sylibr |
⊢ ( 𝑁 ∈ ω → ( ( Fmla ‘ suc 𝑁 ) ∩ { 𝑥 ∣ ( ∃ 𝑢 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑢 ) ∨ ∃ 𝑢 ∈ ( Fmla ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑢 ⊼𝑔 𝑣 ) ) } ) = ∅ ) |