Step |
Hyp |
Ref |
Expression |
1 |
|
goaln0 |
⊢ ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ 𝑁 ) → 𝑁 ≠ ∅ ) |
2 |
1
|
adantl |
⊢ ( ( 𝑁 ∈ ω ∧ ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ 𝑁 ) ) → 𝑁 ≠ ∅ ) |
3 |
|
nnsuc |
⊢ ( ( 𝑁 ∈ ω ∧ 𝑁 ≠ ∅ ) → ∃ 𝑛 ∈ ω 𝑁 = suc 𝑛 ) |
4 |
|
suceq |
⊢ ( 𝑥 = ∅ → suc 𝑥 = suc ∅ ) |
5 |
4
|
fveq2d |
⊢ ( 𝑥 = ∅ → ( Fmla ‘ suc 𝑥 ) = ( Fmla ‘ suc ∅ ) ) |
6 |
5
|
eleq2d |
⊢ ( 𝑥 = ∅ → ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ suc 𝑥 ) ↔ ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ suc ∅ ) ) ) |
7 |
5
|
eleq2d |
⊢ ( 𝑥 = ∅ → ( 𝑎 ∈ ( Fmla ‘ suc 𝑥 ) ↔ 𝑎 ∈ ( Fmla ‘ suc ∅ ) ) ) |
8 |
6 7
|
imbi12d |
⊢ ( 𝑥 = ∅ → ( ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ suc 𝑥 ) → 𝑎 ∈ ( Fmla ‘ suc 𝑥 ) ) ↔ ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ suc ∅ ) → 𝑎 ∈ ( Fmla ‘ suc ∅ ) ) ) ) |
9 |
|
suceq |
⊢ ( 𝑥 = 𝑦 → suc 𝑥 = suc 𝑦 ) |
10 |
9
|
fveq2d |
⊢ ( 𝑥 = 𝑦 → ( Fmla ‘ suc 𝑥 ) = ( Fmla ‘ suc 𝑦 ) ) |
11 |
10
|
eleq2d |
⊢ ( 𝑥 = 𝑦 → ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ suc 𝑥 ) ↔ ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ suc 𝑦 ) ) ) |
12 |
10
|
eleq2d |
⊢ ( 𝑥 = 𝑦 → ( 𝑎 ∈ ( Fmla ‘ suc 𝑥 ) ↔ 𝑎 ∈ ( Fmla ‘ suc 𝑦 ) ) ) |
13 |
11 12
|
imbi12d |
⊢ ( 𝑥 = 𝑦 → ( ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ suc 𝑥 ) → 𝑎 ∈ ( Fmla ‘ suc 𝑥 ) ) ↔ ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ suc 𝑦 ) → 𝑎 ∈ ( Fmla ‘ suc 𝑦 ) ) ) ) |
14 |
|
suceq |
⊢ ( 𝑥 = suc 𝑦 → suc 𝑥 = suc suc 𝑦 ) |
15 |
14
|
fveq2d |
⊢ ( 𝑥 = suc 𝑦 → ( Fmla ‘ suc 𝑥 ) = ( Fmla ‘ suc suc 𝑦 ) ) |
16 |
15
|
eleq2d |
⊢ ( 𝑥 = suc 𝑦 → ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ suc 𝑥 ) ↔ ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ suc suc 𝑦 ) ) ) |
17 |
15
|
eleq2d |
⊢ ( 𝑥 = suc 𝑦 → ( 𝑎 ∈ ( Fmla ‘ suc 𝑥 ) ↔ 𝑎 ∈ ( Fmla ‘ suc suc 𝑦 ) ) ) |
18 |
16 17
|
imbi12d |
⊢ ( 𝑥 = suc 𝑦 → ( ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ suc 𝑥 ) → 𝑎 ∈ ( Fmla ‘ suc 𝑥 ) ) ↔ ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ suc suc 𝑦 ) → 𝑎 ∈ ( Fmla ‘ suc suc 𝑦 ) ) ) ) |
19 |
|
suceq |
⊢ ( 𝑥 = 𝑛 → suc 𝑥 = suc 𝑛 ) |
20 |
19
|
fveq2d |
⊢ ( 𝑥 = 𝑛 → ( Fmla ‘ suc 𝑥 ) = ( Fmla ‘ suc 𝑛 ) ) |
21 |
20
|
eleq2d |
⊢ ( 𝑥 = 𝑛 → ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ suc 𝑥 ) ↔ ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ suc 𝑛 ) ) ) |
22 |
20
|
eleq2d |
⊢ ( 𝑥 = 𝑛 → ( 𝑎 ∈ ( Fmla ‘ suc 𝑥 ) ↔ 𝑎 ∈ ( Fmla ‘ suc 𝑛 ) ) ) |
23 |
21 22
|
imbi12d |
⊢ ( 𝑥 = 𝑛 → ( ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ suc 𝑥 ) → 𝑎 ∈ ( Fmla ‘ suc 𝑥 ) ) ↔ ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ suc 𝑛 ) → 𝑎 ∈ ( Fmla ‘ suc 𝑛 ) ) ) ) |
24 |
|
peano1 |
⊢ ∅ ∈ ω |
25 |
|
df-goal |
⊢ ∀𝑔 𝑖 𝑎 = 〈 2o , 〈 𝑖 , 𝑎 〉 〉 |
26 |
|
opex |
⊢ 〈 2o , 〈 𝑖 , 𝑎 〉 〉 ∈ V |
27 |
25 26
|
eqeltri |
⊢ ∀𝑔 𝑖 𝑎 ∈ V |
28 |
|
isfmlasuc |
⊢ ( ( ∅ ∈ ω ∧ ∀𝑔 𝑖 𝑎 ∈ V ) → ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ suc ∅ ) ↔ ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ ∅ ) ∨ ∃ 𝑢 ∈ ( Fmla ‘ ∅ ) ( ∃ 𝑣 ∈ ( Fmla ‘ ∅ ) ∀𝑔 𝑖 𝑎 = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑘 ∈ ω ∀𝑔 𝑖 𝑎 = ∀𝑔 𝑘 𝑢 ) ) ) ) |
29 |
24 27 28
|
mp2an |
⊢ ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ suc ∅ ) ↔ ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ ∅ ) ∨ ∃ 𝑢 ∈ ( Fmla ‘ ∅ ) ( ∃ 𝑣 ∈ ( Fmla ‘ ∅ ) ∀𝑔 𝑖 𝑎 = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑘 ∈ ω ∀𝑔 𝑖 𝑎 = ∀𝑔 𝑘 𝑢 ) ) ) |
30 |
|
eqeq1 |
⊢ ( 𝑥 = ∀𝑔 𝑖 𝑎 → ( 𝑥 = ( 𝑘 ∈𝑔 𝑗 ) ↔ ∀𝑔 𝑖 𝑎 = ( 𝑘 ∈𝑔 𝑗 ) ) ) |
31 |
30
|
2rexbidv |
⊢ ( 𝑥 = ∀𝑔 𝑖 𝑎 → ( ∃ 𝑘 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑘 ∈𝑔 𝑗 ) ↔ ∃ 𝑘 ∈ ω ∃ 𝑗 ∈ ω ∀𝑔 𝑖 𝑎 = ( 𝑘 ∈𝑔 𝑗 ) ) ) |
32 |
|
fmla0 |
⊢ ( Fmla ‘ ∅ ) = { 𝑥 ∈ V ∣ ∃ 𝑘 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑘 ∈𝑔 𝑗 ) } |
33 |
31 32
|
elrab2 |
⊢ ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ ∅ ) ↔ ( ∀𝑔 𝑖 𝑎 ∈ V ∧ ∃ 𝑘 ∈ ω ∃ 𝑗 ∈ ω ∀𝑔 𝑖 𝑎 = ( 𝑘 ∈𝑔 𝑗 ) ) ) |
34 |
25
|
a1i |
⊢ ( ( 𝑘 ∈ ω ∧ 𝑗 ∈ ω ) → ∀𝑔 𝑖 𝑎 = 〈 2o , 〈 𝑖 , 𝑎 〉 〉 ) |
35 |
|
goel |
⊢ ( ( 𝑘 ∈ ω ∧ 𝑗 ∈ ω ) → ( 𝑘 ∈𝑔 𝑗 ) = 〈 ∅ , 〈 𝑘 , 𝑗 〉 〉 ) |
36 |
34 35
|
eqeq12d |
⊢ ( ( 𝑘 ∈ ω ∧ 𝑗 ∈ ω ) → ( ∀𝑔 𝑖 𝑎 = ( 𝑘 ∈𝑔 𝑗 ) ↔ 〈 2o , 〈 𝑖 , 𝑎 〉 〉 = 〈 ∅ , 〈 𝑘 , 𝑗 〉 〉 ) ) |
37 |
|
2oex |
⊢ 2o ∈ V |
38 |
|
opex |
⊢ 〈 𝑖 , 𝑎 〉 ∈ V |
39 |
37 38
|
opth |
⊢ ( 〈 2o , 〈 𝑖 , 𝑎 〉 〉 = 〈 ∅ , 〈 𝑘 , 𝑗 〉 〉 ↔ ( 2o = ∅ ∧ 〈 𝑖 , 𝑎 〉 = 〈 𝑘 , 𝑗 〉 ) ) |
40 |
|
2on0 |
⊢ 2o ≠ ∅ |
41 |
|
eqneqall |
⊢ ( 2o = ∅ → ( 2o ≠ ∅ → 𝑎 ∈ ( Fmla ‘ suc ∅ ) ) ) |
42 |
40 41
|
mpi |
⊢ ( 2o = ∅ → 𝑎 ∈ ( Fmla ‘ suc ∅ ) ) |
43 |
42
|
adantr |
⊢ ( ( 2o = ∅ ∧ 〈 𝑖 , 𝑎 〉 = 〈 𝑘 , 𝑗 〉 ) → 𝑎 ∈ ( Fmla ‘ suc ∅ ) ) |
44 |
39 43
|
sylbi |
⊢ ( 〈 2o , 〈 𝑖 , 𝑎 〉 〉 = 〈 ∅ , 〈 𝑘 , 𝑗 〉 〉 → 𝑎 ∈ ( Fmla ‘ suc ∅ ) ) |
45 |
36 44
|
syl6bi |
⊢ ( ( 𝑘 ∈ ω ∧ 𝑗 ∈ ω ) → ( ∀𝑔 𝑖 𝑎 = ( 𝑘 ∈𝑔 𝑗 ) → 𝑎 ∈ ( Fmla ‘ suc ∅ ) ) ) |
46 |
45
|
rexlimdva |
⊢ ( 𝑘 ∈ ω → ( ∃ 𝑗 ∈ ω ∀𝑔 𝑖 𝑎 = ( 𝑘 ∈𝑔 𝑗 ) → 𝑎 ∈ ( Fmla ‘ suc ∅ ) ) ) |
47 |
46
|
rexlimiv |
⊢ ( ∃ 𝑘 ∈ ω ∃ 𝑗 ∈ ω ∀𝑔 𝑖 𝑎 = ( 𝑘 ∈𝑔 𝑗 ) → 𝑎 ∈ ( Fmla ‘ suc ∅ ) ) |
48 |
33 47
|
simplbiim |
⊢ ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ ∅ ) → 𝑎 ∈ ( Fmla ‘ suc ∅ ) ) |
49 |
|
gonanegoal |
⊢ ( 𝑢 ⊼𝑔 𝑣 ) ≠ ∀𝑔 𝑖 𝑎 |
50 |
|
eqneqall |
⊢ ( ( 𝑢 ⊼𝑔 𝑣 ) = ∀𝑔 𝑖 𝑎 → ( ( 𝑢 ⊼𝑔 𝑣 ) ≠ ∀𝑔 𝑖 𝑎 → 𝑎 ∈ ( Fmla ‘ suc ∅ ) ) ) |
51 |
49 50
|
mpi |
⊢ ( ( 𝑢 ⊼𝑔 𝑣 ) = ∀𝑔 𝑖 𝑎 → 𝑎 ∈ ( Fmla ‘ suc ∅ ) ) |
52 |
51
|
eqcoms |
⊢ ( ∀𝑔 𝑖 𝑎 = ( 𝑢 ⊼𝑔 𝑣 ) → 𝑎 ∈ ( Fmla ‘ suc ∅ ) ) |
53 |
52
|
a1i |
⊢ ( ( 𝑢 ∈ ( Fmla ‘ ∅ ) ∧ 𝑣 ∈ ( Fmla ‘ ∅ ) ) → ( ∀𝑔 𝑖 𝑎 = ( 𝑢 ⊼𝑔 𝑣 ) → 𝑎 ∈ ( Fmla ‘ suc ∅ ) ) ) |
54 |
53
|
rexlimdva |
⊢ ( 𝑢 ∈ ( Fmla ‘ ∅ ) → ( ∃ 𝑣 ∈ ( Fmla ‘ ∅ ) ∀𝑔 𝑖 𝑎 = ( 𝑢 ⊼𝑔 𝑣 ) → 𝑎 ∈ ( Fmla ‘ suc ∅ ) ) ) |
55 |
|
df-goal |
⊢ ∀𝑔 𝑘 𝑢 = 〈 2o , 〈 𝑘 , 𝑢 〉 〉 |
56 |
25 55
|
eqeq12i |
⊢ ( ∀𝑔 𝑖 𝑎 = ∀𝑔 𝑘 𝑢 ↔ 〈 2o , 〈 𝑖 , 𝑎 〉 〉 = 〈 2o , 〈 𝑘 , 𝑢 〉 〉 ) |
57 |
37 38
|
opth |
⊢ ( 〈 2o , 〈 𝑖 , 𝑎 〉 〉 = 〈 2o , 〈 𝑘 , 𝑢 〉 〉 ↔ ( 2o = 2o ∧ 〈 𝑖 , 𝑎 〉 = 〈 𝑘 , 𝑢 〉 ) ) |
58 |
|
vex |
⊢ 𝑖 ∈ V |
59 |
|
vex |
⊢ 𝑎 ∈ V |
60 |
58 59
|
opth |
⊢ ( 〈 𝑖 , 𝑎 〉 = 〈 𝑘 , 𝑢 〉 ↔ ( 𝑖 = 𝑘 ∧ 𝑎 = 𝑢 ) ) |
61 |
|
eleq1w |
⊢ ( 𝑢 = 𝑎 → ( 𝑢 ∈ ( Fmla ‘ ∅ ) ↔ 𝑎 ∈ ( Fmla ‘ ∅ ) ) ) |
62 |
|
fmlasssuc |
⊢ ( ∅ ∈ ω → ( Fmla ‘ ∅ ) ⊆ ( Fmla ‘ suc ∅ ) ) |
63 |
24 62
|
ax-mp |
⊢ ( Fmla ‘ ∅ ) ⊆ ( Fmla ‘ suc ∅ ) |
64 |
63
|
sseli |
⊢ ( 𝑎 ∈ ( Fmla ‘ ∅ ) → 𝑎 ∈ ( Fmla ‘ suc ∅ ) ) |
65 |
61 64
|
syl6bi |
⊢ ( 𝑢 = 𝑎 → ( 𝑢 ∈ ( Fmla ‘ ∅ ) → 𝑎 ∈ ( Fmla ‘ suc ∅ ) ) ) |
66 |
65
|
eqcoms |
⊢ ( 𝑎 = 𝑢 → ( 𝑢 ∈ ( Fmla ‘ ∅ ) → 𝑎 ∈ ( Fmla ‘ suc ∅ ) ) ) |
67 |
60 66
|
simplbiim |
⊢ ( 〈 𝑖 , 𝑎 〉 = 〈 𝑘 , 𝑢 〉 → ( 𝑢 ∈ ( Fmla ‘ ∅ ) → 𝑎 ∈ ( Fmla ‘ suc ∅ ) ) ) |
68 |
57 67
|
simplbiim |
⊢ ( 〈 2o , 〈 𝑖 , 𝑎 〉 〉 = 〈 2o , 〈 𝑘 , 𝑢 〉 〉 → ( 𝑢 ∈ ( Fmla ‘ ∅ ) → 𝑎 ∈ ( Fmla ‘ suc ∅ ) ) ) |
69 |
68
|
com12 |
⊢ ( 𝑢 ∈ ( Fmla ‘ ∅ ) → ( 〈 2o , 〈 𝑖 , 𝑎 〉 〉 = 〈 2o , 〈 𝑘 , 𝑢 〉 〉 → 𝑎 ∈ ( Fmla ‘ suc ∅ ) ) ) |
70 |
69
|
adantr |
⊢ ( ( 𝑢 ∈ ( Fmla ‘ ∅ ) ∧ 𝑘 ∈ ω ) → ( 〈 2o , 〈 𝑖 , 𝑎 〉 〉 = 〈 2o , 〈 𝑘 , 𝑢 〉 〉 → 𝑎 ∈ ( Fmla ‘ suc ∅ ) ) ) |
71 |
56 70
|
syl5bi |
⊢ ( ( 𝑢 ∈ ( Fmla ‘ ∅ ) ∧ 𝑘 ∈ ω ) → ( ∀𝑔 𝑖 𝑎 = ∀𝑔 𝑘 𝑢 → 𝑎 ∈ ( Fmla ‘ suc ∅ ) ) ) |
72 |
71
|
rexlimdva |
⊢ ( 𝑢 ∈ ( Fmla ‘ ∅ ) → ( ∃ 𝑘 ∈ ω ∀𝑔 𝑖 𝑎 = ∀𝑔 𝑘 𝑢 → 𝑎 ∈ ( Fmla ‘ suc ∅ ) ) ) |
73 |
54 72
|
jaod |
⊢ ( 𝑢 ∈ ( Fmla ‘ ∅ ) → ( ( ∃ 𝑣 ∈ ( Fmla ‘ ∅ ) ∀𝑔 𝑖 𝑎 = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑘 ∈ ω ∀𝑔 𝑖 𝑎 = ∀𝑔 𝑘 𝑢 ) → 𝑎 ∈ ( Fmla ‘ suc ∅ ) ) ) |
74 |
73
|
rexlimiv |
⊢ ( ∃ 𝑢 ∈ ( Fmla ‘ ∅ ) ( ∃ 𝑣 ∈ ( Fmla ‘ ∅ ) ∀𝑔 𝑖 𝑎 = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑘 ∈ ω ∀𝑔 𝑖 𝑎 = ∀𝑔 𝑘 𝑢 ) → 𝑎 ∈ ( Fmla ‘ suc ∅ ) ) |
75 |
48 74
|
jaoi |
⊢ ( ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ ∅ ) ∨ ∃ 𝑢 ∈ ( Fmla ‘ ∅ ) ( ∃ 𝑣 ∈ ( Fmla ‘ ∅ ) ∀𝑔 𝑖 𝑎 = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑘 ∈ ω ∀𝑔 𝑖 𝑎 = ∀𝑔 𝑘 𝑢 ) ) → 𝑎 ∈ ( Fmla ‘ suc ∅ ) ) |
76 |
29 75
|
sylbi |
⊢ ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ suc ∅ ) → 𝑎 ∈ ( Fmla ‘ suc ∅ ) ) |
77 |
|
goalrlem |
⊢ ( 𝑦 ∈ ω → ( ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ suc 𝑦 ) → 𝑎 ∈ ( Fmla ‘ suc 𝑦 ) ) → ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ suc suc 𝑦 ) → 𝑎 ∈ ( Fmla ‘ suc suc 𝑦 ) ) ) ) |
78 |
8 13 18 23 76 77
|
finds |
⊢ ( 𝑛 ∈ ω → ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ suc 𝑛 ) → 𝑎 ∈ ( Fmla ‘ suc 𝑛 ) ) ) |
79 |
78
|
adantr |
⊢ ( ( 𝑛 ∈ ω ∧ 𝑁 = suc 𝑛 ) → ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ suc 𝑛 ) → 𝑎 ∈ ( Fmla ‘ suc 𝑛 ) ) ) |
80 |
|
fveq2 |
⊢ ( 𝑁 = suc 𝑛 → ( Fmla ‘ 𝑁 ) = ( Fmla ‘ suc 𝑛 ) ) |
81 |
80
|
eleq2d |
⊢ ( 𝑁 = suc 𝑛 → ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ 𝑁 ) ↔ ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ suc 𝑛 ) ) ) |
82 |
80
|
eleq2d |
⊢ ( 𝑁 = suc 𝑛 → ( 𝑎 ∈ ( Fmla ‘ 𝑁 ) ↔ 𝑎 ∈ ( Fmla ‘ suc 𝑛 ) ) ) |
83 |
81 82
|
imbi12d |
⊢ ( 𝑁 = suc 𝑛 → ( ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ 𝑁 ) → 𝑎 ∈ ( Fmla ‘ 𝑁 ) ) ↔ ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ suc 𝑛 ) → 𝑎 ∈ ( Fmla ‘ suc 𝑛 ) ) ) ) |
84 |
83
|
adantl |
⊢ ( ( 𝑛 ∈ ω ∧ 𝑁 = suc 𝑛 ) → ( ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ 𝑁 ) → 𝑎 ∈ ( Fmla ‘ 𝑁 ) ) ↔ ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ suc 𝑛 ) → 𝑎 ∈ ( Fmla ‘ suc 𝑛 ) ) ) ) |
85 |
79 84
|
mpbird |
⊢ ( ( 𝑛 ∈ ω ∧ 𝑁 = suc 𝑛 ) → ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ 𝑁 ) → 𝑎 ∈ ( Fmla ‘ 𝑁 ) ) ) |
86 |
85
|
rexlimiva |
⊢ ( ∃ 𝑛 ∈ ω 𝑁 = suc 𝑛 → ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ 𝑁 ) → 𝑎 ∈ ( Fmla ‘ 𝑁 ) ) ) |
87 |
3 86
|
syl |
⊢ ( ( 𝑁 ∈ ω ∧ 𝑁 ≠ ∅ ) → ( ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ 𝑁 ) → 𝑎 ∈ ( Fmla ‘ 𝑁 ) ) ) |
88 |
87
|
impancom |
⊢ ( ( 𝑁 ∈ ω ∧ ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ 𝑁 ) ) → ( 𝑁 ≠ ∅ → 𝑎 ∈ ( Fmla ‘ 𝑁 ) ) ) |
89 |
2 88
|
mpd |
⊢ ( ( 𝑁 ∈ ω ∧ ∀𝑔 𝑖 𝑎 ∈ ( Fmla ‘ 𝑁 ) ) → 𝑎 ∈ ( Fmla ‘ 𝑁 ) ) |