Step |
Hyp |
Ref |
Expression |
1 |
|
gonan0 |
⊢ ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ 𝑁 ) → 𝑁 ≠ ∅ ) |
2 |
1
|
adantl |
⊢ ( ( 𝑁 ∈ ω ∧ ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ 𝑁 ) ) → 𝑁 ≠ ∅ ) |
3 |
|
nnsuc |
⊢ ( ( 𝑁 ∈ ω ∧ 𝑁 ≠ ∅ ) → ∃ 𝑥 ∈ ω 𝑁 = suc 𝑥 ) |
4 |
|
suceq |
⊢ ( 𝑑 = ∅ → suc 𝑑 = suc ∅ ) |
5 |
4
|
fveq2d |
⊢ ( 𝑑 = ∅ → ( Fmla ‘ suc 𝑑 ) = ( Fmla ‘ suc ∅ ) ) |
6 |
5
|
eleq2d |
⊢ ( 𝑑 = ∅ → ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ suc 𝑑 ) ↔ ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ suc ∅ ) ) ) |
7 |
5
|
eleq2d |
⊢ ( 𝑑 = ∅ → ( 𝑎 ∈ ( Fmla ‘ suc 𝑑 ) ↔ 𝑎 ∈ ( Fmla ‘ suc ∅ ) ) ) |
8 |
5
|
eleq2d |
⊢ ( 𝑑 = ∅ → ( 𝑏 ∈ ( Fmla ‘ suc 𝑑 ) ↔ 𝑏 ∈ ( Fmla ‘ suc ∅ ) ) ) |
9 |
7 8
|
anbi12d |
⊢ ( 𝑑 = ∅ → ( ( 𝑎 ∈ ( Fmla ‘ suc 𝑑 ) ∧ 𝑏 ∈ ( Fmla ‘ suc 𝑑 ) ) ↔ ( 𝑎 ∈ ( Fmla ‘ suc ∅ ) ∧ 𝑏 ∈ ( Fmla ‘ suc ∅ ) ) ) ) |
10 |
6 9
|
imbi12d |
⊢ ( 𝑑 = ∅ → ( ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ suc 𝑑 ) → ( 𝑎 ∈ ( Fmla ‘ suc 𝑑 ) ∧ 𝑏 ∈ ( Fmla ‘ suc 𝑑 ) ) ) ↔ ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ suc ∅ ) → ( 𝑎 ∈ ( Fmla ‘ suc ∅ ) ∧ 𝑏 ∈ ( Fmla ‘ suc ∅ ) ) ) ) ) |
11 |
|
suceq |
⊢ ( 𝑑 = 𝑐 → suc 𝑑 = suc 𝑐 ) |
12 |
11
|
fveq2d |
⊢ ( 𝑑 = 𝑐 → ( Fmla ‘ suc 𝑑 ) = ( Fmla ‘ suc 𝑐 ) ) |
13 |
12
|
eleq2d |
⊢ ( 𝑑 = 𝑐 → ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ suc 𝑑 ) ↔ ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ suc 𝑐 ) ) ) |
14 |
12
|
eleq2d |
⊢ ( 𝑑 = 𝑐 → ( 𝑎 ∈ ( Fmla ‘ suc 𝑑 ) ↔ 𝑎 ∈ ( Fmla ‘ suc 𝑐 ) ) ) |
15 |
12
|
eleq2d |
⊢ ( 𝑑 = 𝑐 → ( 𝑏 ∈ ( Fmla ‘ suc 𝑑 ) ↔ 𝑏 ∈ ( Fmla ‘ suc 𝑐 ) ) ) |
16 |
14 15
|
anbi12d |
⊢ ( 𝑑 = 𝑐 → ( ( 𝑎 ∈ ( Fmla ‘ suc 𝑑 ) ∧ 𝑏 ∈ ( Fmla ‘ suc 𝑑 ) ) ↔ ( 𝑎 ∈ ( Fmla ‘ suc 𝑐 ) ∧ 𝑏 ∈ ( Fmla ‘ suc 𝑐 ) ) ) ) |
17 |
13 16
|
imbi12d |
⊢ ( 𝑑 = 𝑐 → ( ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ suc 𝑑 ) → ( 𝑎 ∈ ( Fmla ‘ suc 𝑑 ) ∧ 𝑏 ∈ ( Fmla ‘ suc 𝑑 ) ) ) ↔ ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ suc 𝑐 ) → ( 𝑎 ∈ ( Fmla ‘ suc 𝑐 ) ∧ 𝑏 ∈ ( Fmla ‘ suc 𝑐 ) ) ) ) ) |
18 |
|
suceq |
⊢ ( 𝑑 = suc 𝑐 → suc 𝑑 = suc suc 𝑐 ) |
19 |
18
|
fveq2d |
⊢ ( 𝑑 = suc 𝑐 → ( Fmla ‘ suc 𝑑 ) = ( Fmla ‘ suc suc 𝑐 ) ) |
20 |
19
|
eleq2d |
⊢ ( 𝑑 = suc 𝑐 → ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ suc 𝑑 ) ↔ ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ suc suc 𝑐 ) ) ) |
21 |
19
|
eleq2d |
⊢ ( 𝑑 = suc 𝑐 → ( 𝑎 ∈ ( Fmla ‘ suc 𝑑 ) ↔ 𝑎 ∈ ( Fmla ‘ suc suc 𝑐 ) ) ) |
22 |
19
|
eleq2d |
⊢ ( 𝑑 = suc 𝑐 → ( 𝑏 ∈ ( Fmla ‘ suc 𝑑 ) ↔ 𝑏 ∈ ( Fmla ‘ suc suc 𝑐 ) ) ) |
23 |
21 22
|
anbi12d |
⊢ ( 𝑑 = suc 𝑐 → ( ( 𝑎 ∈ ( Fmla ‘ suc 𝑑 ) ∧ 𝑏 ∈ ( Fmla ‘ suc 𝑑 ) ) ↔ ( 𝑎 ∈ ( Fmla ‘ suc suc 𝑐 ) ∧ 𝑏 ∈ ( Fmla ‘ suc suc 𝑐 ) ) ) ) |
24 |
20 23
|
imbi12d |
⊢ ( 𝑑 = suc 𝑐 → ( ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ suc 𝑑 ) → ( 𝑎 ∈ ( Fmla ‘ suc 𝑑 ) ∧ 𝑏 ∈ ( Fmla ‘ suc 𝑑 ) ) ) ↔ ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ suc suc 𝑐 ) → ( 𝑎 ∈ ( Fmla ‘ suc suc 𝑐 ) ∧ 𝑏 ∈ ( Fmla ‘ suc suc 𝑐 ) ) ) ) ) |
25 |
|
suceq |
⊢ ( 𝑑 = 𝑥 → suc 𝑑 = suc 𝑥 ) |
26 |
25
|
fveq2d |
⊢ ( 𝑑 = 𝑥 → ( Fmla ‘ suc 𝑑 ) = ( Fmla ‘ suc 𝑥 ) ) |
27 |
26
|
eleq2d |
⊢ ( 𝑑 = 𝑥 → ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ suc 𝑑 ) ↔ ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ suc 𝑥 ) ) ) |
28 |
26
|
eleq2d |
⊢ ( 𝑑 = 𝑥 → ( 𝑎 ∈ ( Fmla ‘ suc 𝑑 ) ↔ 𝑎 ∈ ( Fmla ‘ suc 𝑥 ) ) ) |
29 |
26
|
eleq2d |
⊢ ( 𝑑 = 𝑥 → ( 𝑏 ∈ ( Fmla ‘ suc 𝑑 ) ↔ 𝑏 ∈ ( Fmla ‘ suc 𝑥 ) ) ) |
30 |
28 29
|
anbi12d |
⊢ ( 𝑑 = 𝑥 → ( ( 𝑎 ∈ ( Fmla ‘ suc 𝑑 ) ∧ 𝑏 ∈ ( Fmla ‘ suc 𝑑 ) ) ↔ ( 𝑎 ∈ ( Fmla ‘ suc 𝑥 ) ∧ 𝑏 ∈ ( Fmla ‘ suc 𝑥 ) ) ) ) |
31 |
27 30
|
imbi12d |
⊢ ( 𝑑 = 𝑥 → ( ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ suc 𝑑 ) → ( 𝑎 ∈ ( Fmla ‘ suc 𝑑 ) ∧ 𝑏 ∈ ( Fmla ‘ suc 𝑑 ) ) ) ↔ ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ suc 𝑥 ) → ( 𝑎 ∈ ( Fmla ‘ suc 𝑥 ) ∧ 𝑏 ∈ ( Fmla ‘ suc 𝑥 ) ) ) ) ) |
32 |
|
peano1 |
⊢ ∅ ∈ ω |
33 |
|
ovex |
⊢ ( 𝑎 ⊼𝑔 𝑏 ) ∈ V |
34 |
|
isfmlasuc |
⊢ ( ( ∅ ∈ ω ∧ ( 𝑎 ⊼𝑔 𝑏 ) ∈ V ) → ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ suc ∅ ) ↔ ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ ∅ ) ∨ ∃ 𝑢 ∈ ( Fmla ‘ ∅ ) ( ∃ 𝑣 ∈ ( Fmla ‘ ∅ ) ( 𝑎 ⊼𝑔 𝑏 ) = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω ( 𝑎 ⊼𝑔 𝑏 ) = ∀𝑔 𝑖 𝑢 ) ) ) ) |
35 |
32 33 34
|
mp2an |
⊢ ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ suc ∅ ) ↔ ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ ∅ ) ∨ ∃ 𝑢 ∈ ( Fmla ‘ ∅ ) ( ∃ 𝑣 ∈ ( Fmla ‘ ∅ ) ( 𝑎 ⊼𝑔 𝑏 ) = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω ( 𝑎 ⊼𝑔 𝑏 ) = ∀𝑔 𝑖 𝑢 ) ) ) |
36 |
|
eqeq1 |
⊢ ( 𝑥 = ( 𝑎 ⊼𝑔 𝑏 ) → ( 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ↔ ( 𝑎 ⊼𝑔 𝑏 ) = ( 𝑖 ∈𝑔 𝑗 ) ) ) |
37 |
36
|
2rexbidv |
⊢ ( 𝑥 = ( 𝑎 ⊼𝑔 𝑏 ) → ( ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ↔ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑎 ⊼𝑔 𝑏 ) = ( 𝑖 ∈𝑔 𝑗 ) ) ) |
38 |
|
fmla0 |
⊢ ( Fmla ‘ ∅ ) = { 𝑥 ∈ V ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) } |
39 |
37 38
|
elrab2 |
⊢ ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ ∅ ) ↔ ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ V ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑎 ⊼𝑔 𝑏 ) = ( 𝑖 ∈𝑔 𝑗 ) ) ) |
40 |
|
gonafv |
⊢ ( ( 𝑎 ∈ V ∧ 𝑏 ∈ V ) → ( 𝑎 ⊼𝑔 𝑏 ) = 〈 1o , 〈 𝑎 , 𝑏 〉 〉 ) |
41 |
40
|
el2v |
⊢ ( 𝑎 ⊼𝑔 𝑏 ) = 〈 1o , 〈 𝑎 , 𝑏 〉 〉 |
42 |
41
|
a1i |
⊢ ( ( 𝑖 ∈ ω ∧ 𝑗 ∈ ω ) → ( 𝑎 ⊼𝑔 𝑏 ) = 〈 1o , 〈 𝑎 , 𝑏 〉 〉 ) |
43 |
|
goel |
⊢ ( ( 𝑖 ∈ ω ∧ 𝑗 ∈ ω ) → ( 𝑖 ∈𝑔 𝑗 ) = 〈 ∅ , 〈 𝑖 , 𝑗 〉 〉 ) |
44 |
42 43
|
eqeq12d |
⊢ ( ( 𝑖 ∈ ω ∧ 𝑗 ∈ ω ) → ( ( 𝑎 ⊼𝑔 𝑏 ) = ( 𝑖 ∈𝑔 𝑗 ) ↔ 〈 1o , 〈 𝑎 , 𝑏 〉 〉 = 〈 ∅ , 〈 𝑖 , 𝑗 〉 〉 ) ) |
45 |
|
1oex |
⊢ 1o ∈ V |
46 |
|
opex |
⊢ 〈 𝑎 , 𝑏 〉 ∈ V |
47 |
45 46
|
opth |
⊢ ( 〈 1o , 〈 𝑎 , 𝑏 〉 〉 = 〈 ∅ , 〈 𝑖 , 𝑗 〉 〉 ↔ ( 1o = ∅ ∧ 〈 𝑎 , 𝑏 〉 = 〈 𝑖 , 𝑗 〉 ) ) |
48 |
|
1n0 |
⊢ 1o ≠ ∅ |
49 |
|
eqneqall |
⊢ ( 1o = ∅ → ( 1o ≠ ∅ → ( 𝑎 ∈ ( Fmla ‘ suc ∅ ) ∧ 𝑏 ∈ ( Fmla ‘ suc ∅ ) ) ) ) |
50 |
48 49
|
mpi |
⊢ ( 1o = ∅ → ( 𝑎 ∈ ( Fmla ‘ suc ∅ ) ∧ 𝑏 ∈ ( Fmla ‘ suc ∅ ) ) ) |
51 |
50
|
adantr |
⊢ ( ( 1o = ∅ ∧ 〈 𝑎 , 𝑏 〉 = 〈 𝑖 , 𝑗 〉 ) → ( 𝑎 ∈ ( Fmla ‘ suc ∅ ) ∧ 𝑏 ∈ ( Fmla ‘ suc ∅ ) ) ) |
52 |
47 51
|
sylbi |
⊢ ( 〈 1o , 〈 𝑎 , 𝑏 〉 〉 = 〈 ∅ , 〈 𝑖 , 𝑗 〉 〉 → ( 𝑎 ∈ ( Fmla ‘ suc ∅ ) ∧ 𝑏 ∈ ( Fmla ‘ suc ∅ ) ) ) |
53 |
44 52
|
syl6bi |
⊢ ( ( 𝑖 ∈ ω ∧ 𝑗 ∈ ω ) → ( ( 𝑎 ⊼𝑔 𝑏 ) = ( 𝑖 ∈𝑔 𝑗 ) → ( 𝑎 ∈ ( Fmla ‘ suc ∅ ) ∧ 𝑏 ∈ ( Fmla ‘ suc ∅ ) ) ) ) |
54 |
53
|
rexlimdva |
⊢ ( 𝑖 ∈ ω → ( ∃ 𝑗 ∈ ω ( 𝑎 ⊼𝑔 𝑏 ) = ( 𝑖 ∈𝑔 𝑗 ) → ( 𝑎 ∈ ( Fmla ‘ suc ∅ ) ∧ 𝑏 ∈ ( Fmla ‘ suc ∅ ) ) ) ) |
55 |
54
|
rexlimiv |
⊢ ( ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑎 ⊼𝑔 𝑏 ) = ( 𝑖 ∈𝑔 𝑗 ) → ( 𝑎 ∈ ( Fmla ‘ suc ∅ ) ∧ 𝑏 ∈ ( Fmla ‘ suc ∅ ) ) ) |
56 |
55
|
adantl |
⊢ ( ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ V ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑎 ⊼𝑔 𝑏 ) = ( 𝑖 ∈𝑔 𝑗 ) ) → ( 𝑎 ∈ ( Fmla ‘ suc ∅ ) ∧ 𝑏 ∈ ( Fmla ‘ suc ∅ ) ) ) |
57 |
39 56
|
sylbi |
⊢ ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ ∅ ) → ( 𝑎 ∈ ( Fmla ‘ suc ∅ ) ∧ 𝑏 ∈ ( Fmla ‘ suc ∅ ) ) ) |
58 |
41
|
a1i |
⊢ ( ( 𝑢 ∈ ( Fmla ‘ ∅ ) ∧ 𝑣 ∈ ( Fmla ‘ ∅ ) ) → ( 𝑎 ⊼𝑔 𝑏 ) = 〈 1o , 〈 𝑎 , 𝑏 〉 〉 ) |
59 |
|
gonafv |
⊢ ( ( 𝑢 ∈ ( Fmla ‘ ∅ ) ∧ 𝑣 ∈ ( Fmla ‘ ∅ ) ) → ( 𝑢 ⊼𝑔 𝑣 ) = 〈 1o , 〈 𝑢 , 𝑣 〉 〉 ) |
60 |
58 59
|
eqeq12d |
⊢ ( ( 𝑢 ∈ ( Fmla ‘ ∅ ) ∧ 𝑣 ∈ ( Fmla ‘ ∅ ) ) → ( ( 𝑎 ⊼𝑔 𝑏 ) = ( 𝑢 ⊼𝑔 𝑣 ) ↔ 〈 1o , 〈 𝑎 , 𝑏 〉 〉 = 〈 1o , 〈 𝑢 , 𝑣 〉 〉 ) ) |
61 |
45 46
|
opth |
⊢ ( 〈 1o , 〈 𝑎 , 𝑏 〉 〉 = 〈 1o , 〈 𝑢 , 𝑣 〉 〉 ↔ ( 1o = 1o ∧ 〈 𝑎 , 𝑏 〉 = 〈 𝑢 , 𝑣 〉 ) ) |
62 |
|
vex |
⊢ 𝑎 ∈ V |
63 |
|
vex |
⊢ 𝑏 ∈ V |
64 |
62 63
|
opth |
⊢ ( 〈 𝑎 , 𝑏 〉 = 〈 𝑢 , 𝑣 〉 ↔ ( 𝑎 = 𝑢 ∧ 𝑏 = 𝑣 ) ) |
65 |
|
simpl |
⊢ ( ( 𝑎 = 𝑢 ∧ 𝑏 = 𝑣 ) → 𝑎 = 𝑢 ) |
66 |
65
|
equcomd |
⊢ ( ( 𝑎 = 𝑢 ∧ 𝑏 = 𝑣 ) → 𝑢 = 𝑎 ) |
67 |
66
|
eleq1d |
⊢ ( ( 𝑎 = 𝑢 ∧ 𝑏 = 𝑣 ) → ( 𝑢 ∈ ( Fmla ‘ ∅ ) ↔ 𝑎 ∈ ( Fmla ‘ ∅ ) ) ) |
68 |
|
simpr |
⊢ ( ( 𝑎 = 𝑢 ∧ 𝑏 = 𝑣 ) → 𝑏 = 𝑣 ) |
69 |
68
|
equcomd |
⊢ ( ( 𝑎 = 𝑢 ∧ 𝑏 = 𝑣 ) → 𝑣 = 𝑏 ) |
70 |
69
|
eleq1d |
⊢ ( ( 𝑎 = 𝑢 ∧ 𝑏 = 𝑣 ) → ( 𝑣 ∈ ( Fmla ‘ ∅ ) ↔ 𝑏 ∈ ( Fmla ‘ ∅ ) ) ) |
71 |
67 70
|
anbi12d |
⊢ ( ( 𝑎 = 𝑢 ∧ 𝑏 = 𝑣 ) → ( ( 𝑢 ∈ ( Fmla ‘ ∅ ) ∧ 𝑣 ∈ ( Fmla ‘ ∅ ) ) ↔ ( 𝑎 ∈ ( Fmla ‘ ∅ ) ∧ 𝑏 ∈ ( Fmla ‘ ∅ ) ) ) ) |
72 |
64 71
|
sylbi |
⊢ ( 〈 𝑎 , 𝑏 〉 = 〈 𝑢 , 𝑣 〉 → ( ( 𝑢 ∈ ( Fmla ‘ ∅ ) ∧ 𝑣 ∈ ( Fmla ‘ ∅ ) ) ↔ ( 𝑎 ∈ ( Fmla ‘ ∅ ) ∧ 𝑏 ∈ ( Fmla ‘ ∅ ) ) ) ) |
73 |
72
|
adantl |
⊢ ( ( 1o = 1o ∧ 〈 𝑎 , 𝑏 〉 = 〈 𝑢 , 𝑣 〉 ) → ( ( 𝑢 ∈ ( Fmla ‘ ∅ ) ∧ 𝑣 ∈ ( Fmla ‘ ∅ ) ) ↔ ( 𝑎 ∈ ( Fmla ‘ ∅ ) ∧ 𝑏 ∈ ( Fmla ‘ ∅ ) ) ) ) |
74 |
61 73
|
sylbi |
⊢ ( 〈 1o , 〈 𝑎 , 𝑏 〉 〉 = 〈 1o , 〈 𝑢 , 𝑣 〉 〉 → ( ( 𝑢 ∈ ( Fmla ‘ ∅ ) ∧ 𝑣 ∈ ( Fmla ‘ ∅ ) ) ↔ ( 𝑎 ∈ ( Fmla ‘ ∅ ) ∧ 𝑏 ∈ ( Fmla ‘ ∅ ) ) ) ) |
75 |
|
fmlasssuc |
⊢ ( ∅ ∈ ω → ( Fmla ‘ ∅ ) ⊆ ( Fmla ‘ suc ∅ ) ) |
76 |
32 75
|
ax-mp |
⊢ ( Fmla ‘ ∅ ) ⊆ ( Fmla ‘ suc ∅ ) |
77 |
76
|
sseli |
⊢ ( 𝑎 ∈ ( Fmla ‘ ∅ ) → 𝑎 ∈ ( Fmla ‘ suc ∅ ) ) |
78 |
76
|
sseli |
⊢ ( 𝑏 ∈ ( Fmla ‘ ∅ ) → 𝑏 ∈ ( Fmla ‘ suc ∅ ) ) |
79 |
77 78
|
anim12i |
⊢ ( ( 𝑎 ∈ ( Fmla ‘ ∅ ) ∧ 𝑏 ∈ ( Fmla ‘ ∅ ) ) → ( 𝑎 ∈ ( Fmla ‘ suc ∅ ) ∧ 𝑏 ∈ ( Fmla ‘ suc ∅ ) ) ) |
80 |
74 79
|
syl6bi |
⊢ ( 〈 1o , 〈 𝑎 , 𝑏 〉 〉 = 〈 1o , 〈 𝑢 , 𝑣 〉 〉 → ( ( 𝑢 ∈ ( Fmla ‘ ∅ ) ∧ 𝑣 ∈ ( Fmla ‘ ∅ ) ) → ( 𝑎 ∈ ( Fmla ‘ suc ∅ ) ∧ 𝑏 ∈ ( Fmla ‘ suc ∅ ) ) ) ) |
81 |
80
|
com12 |
⊢ ( ( 𝑢 ∈ ( Fmla ‘ ∅ ) ∧ 𝑣 ∈ ( Fmla ‘ ∅ ) ) → ( 〈 1o , 〈 𝑎 , 𝑏 〉 〉 = 〈 1o , 〈 𝑢 , 𝑣 〉 〉 → ( 𝑎 ∈ ( Fmla ‘ suc ∅ ) ∧ 𝑏 ∈ ( Fmla ‘ suc ∅ ) ) ) ) |
82 |
60 81
|
sylbid |
⊢ ( ( 𝑢 ∈ ( Fmla ‘ ∅ ) ∧ 𝑣 ∈ ( Fmla ‘ ∅ ) ) → ( ( 𝑎 ⊼𝑔 𝑏 ) = ( 𝑢 ⊼𝑔 𝑣 ) → ( 𝑎 ∈ ( Fmla ‘ suc ∅ ) ∧ 𝑏 ∈ ( Fmla ‘ suc ∅ ) ) ) ) |
83 |
82
|
rexlimdva |
⊢ ( 𝑢 ∈ ( Fmla ‘ ∅ ) → ( ∃ 𝑣 ∈ ( Fmla ‘ ∅ ) ( 𝑎 ⊼𝑔 𝑏 ) = ( 𝑢 ⊼𝑔 𝑣 ) → ( 𝑎 ∈ ( Fmla ‘ suc ∅ ) ∧ 𝑏 ∈ ( Fmla ‘ suc ∅ ) ) ) ) |
84 |
|
gonanegoal |
⊢ ( 𝑎 ⊼𝑔 𝑏 ) ≠ ∀𝑔 𝑖 𝑢 |
85 |
|
eqneqall |
⊢ ( ( 𝑎 ⊼𝑔 𝑏 ) = ∀𝑔 𝑖 𝑢 → ( ( 𝑎 ⊼𝑔 𝑏 ) ≠ ∀𝑔 𝑖 𝑢 → ( 𝑎 ∈ ( Fmla ‘ suc ∅ ) ∧ 𝑏 ∈ ( Fmla ‘ suc ∅ ) ) ) ) |
86 |
84 85
|
mpi |
⊢ ( ( 𝑎 ⊼𝑔 𝑏 ) = ∀𝑔 𝑖 𝑢 → ( 𝑎 ∈ ( Fmla ‘ suc ∅ ) ∧ 𝑏 ∈ ( Fmla ‘ suc ∅ ) ) ) |
87 |
86
|
a1i |
⊢ ( ( 𝑢 ∈ ( Fmla ‘ ∅ ) ∧ 𝑖 ∈ ω ) → ( ( 𝑎 ⊼𝑔 𝑏 ) = ∀𝑔 𝑖 𝑢 → ( 𝑎 ∈ ( Fmla ‘ suc ∅ ) ∧ 𝑏 ∈ ( Fmla ‘ suc ∅ ) ) ) ) |
88 |
87
|
rexlimdva |
⊢ ( 𝑢 ∈ ( Fmla ‘ ∅ ) → ( ∃ 𝑖 ∈ ω ( 𝑎 ⊼𝑔 𝑏 ) = ∀𝑔 𝑖 𝑢 → ( 𝑎 ∈ ( Fmla ‘ suc ∅ ) ∧ 𝑏 ∈ ( Fmla ‘ suc ∅ ) ) ) ) |
89 |
83 88
|
jaod |
⊢ ( 𝑢 ∈ ( Fmla ‘ ∅ ) → ( ( ∃ 𝑣 ∈ ( Fmla ‘ ∅ ) ( 𝑎 ⊼𝑔 𝑏 ) = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω ( 𝑎 ⊼𝑔 𝑏 ) = ∀𝑔 𝑖 𝑢 ) → ( 𝑎 ∈ ( Fmla ‘ suc ∅ ) ∧ 𝑏 ∈ ( Fmla ‘ suc ∅ ) ) ) ) |
90 |
89
|
rexlimiv |
⊢ ( ∃ 𝑢 ∈ ( Fmla ‘ ∅ ) ( ∃ 𝑣 ∈ ( Fmla ‘ ∅ ) ( 𝑎 ⊼𝑔 𝑏 ) = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω ( 𝑎 ⊼𝑔 𝑏 ) = ∀𝑔 𝑖 𝑢 ) → ( 𝑎 ∈ ( Fmla ‘ suc ∅ ) ∧ 𝑏 ∈ ( Fmla ‘ suc ∅ ) ) ) |
91 |
57 90
|
jaoi |
⊢ ( ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ ∅ ) ∨ ∃ 𝑢 ∈ ( Fmla ‘ ∅ ) ( ∃ 𝑣 ∈ ( Fmla ‘ ∅ ) ( 𝑎 ⊼𝑔 𝑏 ) = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω ( 𝑎 ⊼𝑔 𝑏 ) = ∀𝑔 𝑖 𝑢 ) ) → ( 𝑎 ∈ ( Fmla ‘ suc ∅ ) ∧ 𝑏 ∈ ( Fmla ‘ suc ∅ ) ) ) |
92 |
35 91
|
sylbi |
⊢ ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ suc ∅ ) → ( 𝑎 ∈ ( Fmla ‘ suc ∅ ) ∧ 𝑏 ∈ ( Fmla ‘ suc ∅ ) ) ) |
93 |
|
gonarlem |
⊢ ( 𝑐 ∈ ω → ( ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ suc 𝑐 ) → ( 𝑎 ∈ ( Fmla ‘ suc 𝑐 ) ∧ 𝑏 ∈ ( Fmla ‘ suc 𝑐 ) ) ) → ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ suc suc 𝑐 ) → ( 𝑎 ∈ ( Fmla ‘ suc suc 𝑐 ) ∧ 𝑏 ∈ ( Fmla ‘ suc suc 𝑐 ) ) ) ) ) |
94 |
10 17 24 31 92 93
|
finds |
⊢ ( 𝑥 ∈ ω → ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ suc 𝑥 ) → ( 𝑎 ∈ ( Fmla ‘ suc 𝑥 ) ∧ 𝑏 ∈ ( Fmla ‘ suc 𝑥 ) ) ) ) |
95 |
94
|
adantr |
⊢ ( ( 𝑥 ∈ ω ∧ 𝑁 = suc 𝑥 ) → ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ suc 𝑥 ) → ( 𝑎 ∈ ( Fmla ‘ suc 𝑥 ) ∧ 𝑏 ∈ ( Fmla ‘ suc 𝑥 ) ) ) ) |
96 |
|
fveq2 |
⊢ ( 𝑁 = suc 𝑥 → ( Fmla ‘ 𝑁 ) = ( Fmla ‘ suc 𝑥 ) ) |
97 |
96
|
eleq2d |
⊢ ( 𝑁 = suc 𝑥 → ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ 𝑁 ) ↔ ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ suc 𝑥 ) ) ) |
98 |
96
|
eleq2d |
⊢ ( 𝑁 = suc 𝑥 → ( 𝑎 ∈ ( Fmla ‘ 𝑁 ) ↔ 𝑎 ∈ ( Fmla ‘ suc 𝑥 ) ) ) |
99 |
96
|
eleq2d |
⊢ ( 𝑁 = suc 𝑥 → ( 𝑏 ∈ ( Fmla ‘ 𝑁 ) ↔ 𝑏 ∈ ( Fmla ‘ suc 𝑥 ) ) ) |
100 |
98 99
|
anbi12d |
⊢ ( 𝑁 = suc 𝑥 → ( ( 𝑎 ∈ ( Fmla ‘ 𝑁 ) ∧ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ) ↔ ( 𝑎 ∈ ( Fmla ‘ suc 𝑥 ) ∧ 𝑏 ∈ ( Fmla ‘ suc 𝑥 ) ) ) ) |
101 |
97 100
|
imbi12d |
⊢ ( 𝑁 = suc 𝑥 → ( ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ 𝑁 ) → ( 𝑎 ∈ ( Fmla ‘ 𝑁 ) ∧ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ) ) ↔ ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ suc 𝑥 ) → ( 𝑎 ∈ ( Fmla ‘ suc 𝑥 ) ∧ 𝑏 ∈ ( Fmla ‘ suc 𝑥 ) ) ) ) ) |
102 |
101
|
adantl |
⊢ ( ( 𝑥 ∈ ω ∧ 𝑁 = suc 𝑥 ) → ( ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ 𝑁 ) → ( 𝑎 ∈ ( Fmla ‘ 𝑁 ) ∧ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ) ) ↔ ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ suc 𝑥 ) → ( 𝑎 ∈ ( Fmla ‘ suc 𝑥 ) ∧ 𝑏 ∈ ( Fmla ‘ suc 𝑥 ) ) ) ) ) |
103 |
95 102
|
mpbird |
⊢ ( ( 𝑥 ∈ ω ∧ 𝑁 = suc 𝑥 ) → ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ 𝑁 ) → ( 𝑎 ∈ ( Fmla ‘ 𝑁 ) ∧ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ) ) ) |
104 |
103
|
rexlimiva |
⊢ ( ∃ 𝑥 ∈ ω 𝑁 = suc 𝑥 → ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ 𝑁 ) → ( 𝑎 ∈ ( Fmla ‘ 𝑁 ) ∧ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ) ) ) |
105 |
3 104
|
syl |
⊢ ( ( 𝑁 ∈ ω ∧ 𝑁 ≠ ∅ ) → ( ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ 𝑁 ) → ( 𝑎 ∈ ( Fmla ‘ 𝑁 ) ∧ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ) ) ) |
106 |
105
|
impancom |
⊢ ( ( 𝑁 ∈ ω ∧ ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ 𝑁 ) ) → ( 𝑁 ≠ ∅ → ( 𝑎 ∈ ( Fmla ‘ 𝑁 ) ∧ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ) ) ) |
107 |
2 106
|
mpd |
⊢ ( ( 𝑁 ∈ ω ∧ ( 𝑎 ⊼𝑔 𝑏 ) ∈ ( Fmla ‘ 𝑁 ) ) → ( 𝑎 ∈ ( Fmla ‘ 𝑁 ) ∧ 𝑏 ∈ ( Fmla ‘ 𝑁 ) ) ) |