Step |
Hyp |
Ref |
Expression |
1 |
|
fveq2 |
⊢ ( 𝑥 = ∅ → ( Fmla ‘ 𝑥 ) = ( Fmla ‘ ∅ ) ) |
2 |
1
|
eleq2d |
⊢ ( 𝑥 = ∅ → ( ∅ ∈ ( Fmla ‘ 𝑥 ) ↔ ∅ ∈ ( Fmla ‘ ∅ ) ) ) |
3 |
2
|
notbid |
⊢ ( 𝑥 = ∅ → ( ¬ ∅ ∈ ( Fmla ‘ 𝑥 ) ↔ ¬ ∅ ∈ ( Fmla ‘ ∅ ) ) ) |
4 |
|
fveq2 |
⊢ ( 𝑥 = 𝑦 → ( Fmla ‘ 𝑥 ) = ( Fmla ‘ 𝑦 ) ) |
5 |
4
|
eleq2d |
⊢ ( 𝑥 = 𝑦 → ( ∅ ∈ ( Fmla ‘ 𝑥 ) ↔ ∅ ∈ ( Fmla ‘ 𝑦 ) ) ) |
6 |
5
|
notbid |
⊢ ( 𝑥 = 𝑦 → ( ¬ ∅ ∈ ( Fmla ‘ 𝑥 ) ↔ ¬ ∅ ∈ ( Fmla ‘ 𝑦 ) ) ) |
7 |
|
fveq2 |
⊢ ( 𝑥 = suc 𝑦 → ( Fmla ‘ 𝑥 ) = ( Fmla ‘ suc 𝑦 ) ) |
8 |
7
|
eleq2d |
⊢ ( 𝑥 = suc 𝑦 → ( ∅ ∈ ( Fmla ‘ 𝑥 ) ↔ ∅ ∈ ( Fmla ‘ suc 𝑦 ) ) ) |
9 |
8
|
notbid |
⊢ ( 𝑥 = suc 𝑦 → ( ¬ ∅ ∈ ( Fmla ‘ 𝑥 ) ↔ ¬ ∅ ∈ ( Fmla ‘ suc 𝑦 ) ) ) |
10 |
|
fveq2 |
⊢ ( 𝑥 = 𝑁 → ( Fmla ‘ 𝑥 ) = ( Fmla ‘ 𝑁 ) ) |
11 |
10
|
eleq2d |
⊢ ( 𝑥 = 𝑁 → ( ∅ ∈ ( Fmla ‘ 𝑥 ) ↔ ∅ ∈ ( Fmla ‘ 𝑁 ) ) ) |
12 |
11
|
notbid |
⊢ ( 𝑥 = 𝑁 → ( ¬ ∅ ∈ ( Fmla ‘ 𝑥 ) ↔ ¬ ∅ ∈ ( Fmla ‘ 𝑁 ) ) ) |
13 |
|
0ex |
⊢ ∅ ∈ V |
14 |
|
opex |
⊢ 〈 𝑖 , 𝑗 〉 ∈ V |
15 |
13 14
|
pm3.2i |
⊢ ( ∅ ∈ V ∧ 〈 𝑖 , 𝑗 〉 ∈ V ) |
16 |
15
|
a1i |
⊢ ( ( 𝑖 ∈ ω ∧ 𝑗 ∈ ω ) → ( ∅ ∈ V ∧ 〈 𝑖 , 𝑗 〉 ∈ V ) ) |
17 |
|
necom |
⊢ ( ∅ ≠ 〈 ∅ , 〈 𝑖 , 𝑗 〉 〉 ↔ 〈 ∅ , 〈 𝑖 , 𝑗 〉 〉 ≠ ∅ ) |
18 |
|
opnz |
⊢ ( 〈 ∅ , 〈 𝑖 , 𝑗 〉 〉 ≠ ∅ ↔ ( ∅ ∈ V ∧ 〈 𝑖 , 𝑗 〉 ∈ V ) ) |
19 |
17 18
|
bitri |
⊢ ( ∅ ≠ 〈 ∅ , 〈 𝑖 , 𝑗 〉 〉 ↔ ( ∅ ∈ V ∧ 〈 𝑖 , 𝑗 〉 ∈ V ) ) |
20 |
16 19
|
sylibr |
⊢ ( ( 𝑖 ∈ ω ∧ 𝑗 ∈ ω ) → ∅ ≠ 〈 ∅ , 〈 𝑖 , 𝑗 〉 〉 ) |
21 |
20
|
neneqd |
⊢ ( ( 𝑖 ∈ ω ∧ 𝑗 ∈ ω ) → ¬ ∅ = 〈 ∅ , 〈 𝑖 , 𝑗 〉 〉 ) |
22 |
|
goel |
⊢ ( ( 𝑖 ∈ ω ∧ 𝑗 ∈ ω ) → ( 𝑖 ∈𝑔 𝑗 ) = 〈 ∅ , 〈 𝑖 , 𝑗 〉 〉 ) |
23 |
22
|
eqeq2d |
⊢ ( ( 𝑖 ∈ ω ∧ 𝑗 ∈ ω ) → ( ∅ = ( 𝑖 ∈𝑔 𝑗 ) ↔ ∅ = 〈 ∅ , 〈 𝑖 , 𝑗 〉 〉 ) ) |
24 |
21 23
|
mtbird |
⊢ ( ( 𝑖 ∈ ω ∧ 𝑗 ∈ ω ) → ¬ ∅ = ( 𝑖 ∈𝑔 𝑗 ) ) |
25 |
24
|
rgen2 |
⊢ ∀ 𝑖 ∈ ω ∀ 𝑗 ∈ ω ¬ ∅ = ( 𝑖 ∈𝑔 𝑗 ) |
26 |
|
ralnex2 |
⊢ ( ∀ 𝑖 ∈ ω ∀ 𝑗 ∈ ω ¬ ∅ = ( 𝑖 ∈𝑔 𝑗 ) ↔ ¬ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ∅ = ( 𝑖 ∈𝑔 𝑗 ) ) |
27 |
25 26
|
mpbi |
⊢ ¬ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ∅ = ( 𝑖 ∈𝑔 𝑗 ) |
28 |
27
|
intnan |
⊢ ¬ ( ∅ ∈ V ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ∅ = ( 𝑖 ∈𝑔 𝑗 ) ) |
29 |
|
fmla0 |
⊢ ( Fmla ‘ ∅ ) = { 𝑥 ∈ V ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) } |
30 |
29
|
eleq2i |
⊢ ( ∅ ∈ ( Fmla ‘ ∅ ) ↔ ∅ ∈ { 𝑥 ∈ V ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) } ) |
31 |
|
eqeq1 |
⊢ ( 𝑥 = ∅ → ( 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ↔ ∅ = ( 𝑖 ∈𝑔 𝑗 ) ) ) |
32 |
31
|
2rexbidv |
⊢ ( 𝑥 = ∅ → ( ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ↔ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ∅ = ( 𝑖 ∈𝑔 𝑗 ) ) ) |
33 |
32
|
elrab |
⊢ ( ∅ ∈ { 𝑥 ∈ V ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) } ↔ ( ∅ ∈ V ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ∅ = ( 𝑖 ∈𝑔 𝑗 ) ) ) |
34 |
30 33
|
bitri |
⊢ ( ∅ ∈ ( Fmla ‘ ∅ ) ↔ ( ∅ ∈ V ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ∅ = ( 𝑖 ∈𝑔 𝑗 ) ) ) |
35 |
28 34
|
mtbir |
⊢ ¬ ∅ ∈ ( Fmla ‘ ∅ ) |
36 |
|
simpr |
⊢ ( ( 𝑦 ∈ ω ∧ ¬ ∅ ∈ ( Fmla ‘ 𝑦 ) ) → ¬ ∅ ∈ ( Fmla ‘ 𝑦 ) ) |
37 |
|
1oex |
⊢ 1o ∈ V |
38 |
|
opex |
⊢ 〈 𝑢 , 𝑣 〉 ∈ V |
39 |
37 38
|
opnzi |
⊢ 〈 1o , 〈 𝑢 , 𝑣 〉 〉 ≠ ∅ |
40 |
39
|
nesymi |
⊢ ¬ ∅ = 〈 1o , 〈 𝑢 , 𝑣 〉 〉 |
41 |
|
gonafv |
⊢ ( ( 𝑢 ∈ ( Fmla ‘ 𝑦 ) ∧ 𝑣 ∈ ( Fmla ‘ 𝑦 ) ) → ( 𝑢 ⊼𝑔 𝑣 ) = 〈 1o , 〈 𝑢 , 𝑣 〉 〉 ) |
42 |
41
|
adantll |
⊢ ( ( ( 𝑦 ∈ ω ∧ 𝑢 ∈ ( Fmla ‘ 𝑦 ) ) ∧ 𝑣 ∈ ( Fmla ‘ 𝑦 ) ) → ( 𝑢 ⊼𝑔 𝑣 ) = 〈 1o , 〈 𝑢 , 𝑣 〉 〉 ) |
43 |
42
|
eqeq2d |
⊢ ( ( ( 𝑦 ∈ ω ∧ 𝑢 ∈ ( Fmla ‘ 𝑦 ) ) ∧ 𝑣 ∈ ( Fmla ‘ 𝑦 ) ) → ( ∅ = ( 𝑢 ⊼𝑔 𝑣 ) ↔ ∅ = 〈 1o , 〈 𝑢 , 𝑣 〉 〉 ) ) |
44 |
40 43
|
mtbiri |
⊢ ( ( ( 𝑦 ∈ ω ∧ 𝑢 ∈ ( Fmla ‘ 𝑦 ) ) ∧ 𝑣 ∈ ( Fmla ‘ 𝑦 ) ) → ¬ ∅ = ( 𝑢 ⊼𝑔 𝑣 ) ) |
45 |
44
|
ralrimiva |
⊢ ( ( 𝑦 ∈ ω ∧ 𝑢 ∈ ( Fmla ‘ 𝑦 ) ) → ∀ 𝑣 ∈ ( Fmla ‘ 𝑦 ) ¬ ∅ = ( 𝑢 ⊼𝑔 𝑣 ) ) |
46 |
|
2oex |
⊢ 2o ∈ V |
47 |
|
opex |
⊢ 〈 𝑖 , 𝑢 〉 ∈ V |
48 |
46 47
|
opnzi |
⊢ 〈 2o , 〈 𝑖 , 𝑢 〉 〉 ≠ ∅ |
49 |
48
|
nesymi |
⊢ ¬ ∅ = 〈 2o , 〈 𝑖 , 𝑢 〉 〉 |
50 |
|
df-goal |
⊢ ∀𝑔 𝑖 𝑢 = 〈 2o , 〈 𝑖 , 𝑢 〉 〉 |
51 |
50
|
eqeq2i |
⊢ ( ∅ = ∀𝑔 𝑖 𝑢 ↔ ∅ = 〈 2o , 〈 𝑖 , 𝑢 〉 〉 ) |
52 |
49 51
|
mtbir |
⊢ ¬ ∅ = ∀𝑔 𝑖 𝑢 |
53 |
52
|
a1i |
⊢ ( ( ( 𝑦 ∈ ω ∧ 𝑢 ∈ ( Fmla ‘ 𝑦 ) ) ∧ 𝑖 ∈ ω ) → ¬ ∅ = ∀𝑔 𝑖 𝑢 ) |
54 |
53
|
ralrimiva |
⊢ ( ( 𝑦 ∈ ω ∧ 𝑢 ∈ ( Fmla ‘ 𝑦 ) ) → ∀ 𝑖 ∈ ω ¬ ∅ = ∀𝑔 𝑖 𝑢 ) |
55 |
45 54
|
jca |
⊢ ( ( 𝑦 ∈ ω ∧ 𝑢 ∈ ( Fmla ‘ 𝑦 ) ) → ( ∀ 𝑣 ∈ ( Fmla ‘ 𝑦 ) ¬ ∅ = ( 𝑢 ⊼𝑔 𝑣 ) ∧ ∀ 𝑖 ∈ ω ¬ ∅ = ∀𝑔 𝑖 𝑢 ) ) |
56 |
55
|
ralrimiva |
⊢ ( 𝑦 ∈ ω → ∀ 𝑢 ∈ ( Fmla ‘ 𝑦 ) ( ∀ 𝑣 ∈ ( Fmla ‘ 𝑦 ) ¬ ∅ = ( 𝑢 ⊼𝑔 𝑣 ) ∧ ∀ 𝑖 ∈ ω ¬ ∅ = ∀𝑔 𝑖 𝑢 ) ) |
57 |
56
|
adantr |
⊢ ( ( 𝑦 ∈ ω ∧ ¬ ∅ ∈ ( Fmla ‘ 𝑦 ) ) → ∀ 𝑢 ∈ ( Fmla ‘ 𝑦 ) ( ∀ 𝑣 ∈ ( Fmla ‘ 𝑦 ) ¬ ∅ = ( 𝑢 ⊼𝑔 𝑣 ) ∧ ∀ 𝑖 ∈ ω ¬ ∅ = ∀𝑔 𝑖 𝑢 ) ) |
58 |
|
ralnex |
⊢ ( ∀ 𝑣 ∈ ( Fmla ‘ 𝑦 ) ¬ ∅ = ( 𝑢 ⊼𝑔 𝑣 ) ↔ ¬ ∃ 𝑣 ∈ ( Fmla ‘ 𝑦 ) ∅ = ( 𝑢 ⊼𝑔 𝑣 ) ) |
59 |
|
ralnex |
⊢ ( ∀ 𝑖 ∈ ω ¬ ∅ = ∀𝑔 𝑖 𝑢 ↔ ¬ ∃ 𝑖 ∈ ω ∅ = ∀𝑔 𝑖 𝑢 ) |
60 |
58 59
|
anbi12i |
⊢ ( ( ∀ 𝑣 ∈ ( Fmla ‘ 𝑦 ) ¬ ∅ = ( 𝑢 ⊼𝑔 𝑣 ) ∧ ∀ 𝑖 ∈ ω ¬ ∅ = ∀𝑔 𝑖 𝑢 ) ↔ ( ¬ ∃ 𝑣 ∈ ( Fmla ‘ 𝑦 ) ∅ = ( 𝑢 ⊼𝑔 𝑣 ) ∧ ¬ ∃ 𝑖 ∈ ω ∅ = ∀𝑔 𝑖 𝑢 ) ) |
61 |
|
ioran |
⊢ ( ¬ ( ∃ 𝑣 ∈ ( Fmla ‘ 𝑦 ) ∅ = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω ∅ = ∀𝑔 𝑖 𝑢 ) ↔ ( ¬ ∃ 𝑣 ∈ ( Fmla ‘ 𝑦 ) ∅ = ( 𝑢 ⊼𝑔 𝑣 ) ∧ ¬ ∃ 𝑖 ∈ ω ∅ = ∀𝑔 𝑖 𝑢 ) ) |
62 |
60 61
|
bitr4i |
⊢ ( ( ∀ 𝑣 ∈ ( Fmla ‘ 𝑦 ) ¬ ∅ = ( 𝑢 ⊼𝑔 𝑣 ) ∧ ∀ 𝑖 ∈ ω ¬ ∅ = ∀𝑔 𝑖 𝑢 ) ↔ ¬ ( ∃ 𝑣 ∈ ( Fmla ‘ 𝑦 ) ∅ = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω ∅ = ∀𝑔 𝑖 𝑢 ) ) |
63 |
62
|
ralbii |
⊢ ( ∀ 𝑢 ∈ ( Fmla ‘ 𝑦 ) ( ∀ 𝑣 ∈ ( Fmla ‘ 𝑦 ) ¬ ∅ = ( 𝑢 ⊼𝑔 𝑣 ) ∧ ∀ 𝑖 ∈ ω ¬ ∅ = ∀𝑔 𝑖 𝑢 ) ↔ ∀ 𝑢 ∈ ( Fmla ‘ 𝑦 ) ¬ ( ∃ 𝑣 ∈ ( Fmla ‘ 𝑦 ) ∅ = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω ∅ = ∀𝑔 𝑖 𝑢 ) ) |
64 |
|
ralnex |
⊢ ( ∀ 𝑢 ∈ ( Fmla ‘ 𝑦 ) ¬ ( ∃ 𝑣 ∈ ( Fmla ‘ 𝑦 ) ∅ = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω ∅ = ∀𝑔 𝑖 𝑢 ) ↔ ¬ ∃ 𝑢 ∈ ( Fmla ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( Fmla ‘ 𝑦 ) ∅ = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω ∅ = ∀𝑔 𝑖 𝑢 ) ) |
65 |
63 64
|
bitri |
⊢ ( ∀ 𝑢 ∈ ( Fmla ‘ 𝑦 ) ( ∀ 𝑣 ∈ ( Fmla ‘ 𝑦 ) ¬ ∅ = ( 𝑢 ⊼𝑔 𝑣 ) ∧ ∀ 𝑖 ∈ ω ¬ ∅ = ∀𝑔 𝑖 𝑢 ) ↔ ¬ ∃ 𝑢 ∈ ( Fmla ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( Fmla ‘ 𝑦 ) ∅ = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω ∅ = ∀𝑔 𝑖 𝑢 ) ) |
66 |
57 65
|
sylib |
⊢ ( ( 𝑦 ∈ ω ∧ ¬ ∅ ∈ ( Fmla ‘ 𝑦 ) ) → ¬ ∃ 𝑢 ∈ ( Fmla ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( Fmla ‘ 𝑦 ) ∅ = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω ∅ = ∀𝑔 𝑖 𝑢 ) ) |
67 |
|
ioran |
⊢ ( ¬ ( ∅ ∈ ( Fmla ‘ 𝑦 ) ∨ ∃ 𝑢 ∈ ( Fmla ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( Fmla ‘ 𝑦 ) ∅ = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω ∅ = ∀𝑔 𝑖 𝑢 ) ) ↔ ( ¬ ∅ ∈ ( Fmla ‘ 𝑦 ) ∧ ¬ ∃ 𝑢 ∈ ( Fmla ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( Fmla ‘ 𝑦 ) ∅ = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω ∅ = ∀𝑔 𝑖 𝑢 ) ) ) |
68 |
36 66 67
|
sylanbrc |
⊢ ( ( 𝑦 ∈ ω ∧ ¬ ∅ ∈ ( Fmla ‘ 𝑦 ) ) → ¬ ( ∅ ∈ ( Fmla ‘ 𝑦 ) ∨ ∃ 𝑢 ∈ ( Fmla ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( Fmla ‘ 𝑦 ) ∅ = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω ∅ = ∀𝑔 𝑖 𝑢 ) ) ) |
69 |
|
fmlasuc |
⊢ ( 𝑦 ∈ ω → ( Fmla ‘ suc 𝑦 ) = ( ( Fmla ‘ 𝑦 ) ∪ { 𝑥 ∣ ∃ 𝑢 ∈ ( Fmla ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( Fmla ‘ 𝑦 ) 𝑥 = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑢 ) } ) ) |
70 |
69
|
eleq2d |
⊢ ( 𝑦 ∈ ω → ( ∅ ∈ ( Fmla ‘ suc 𝑦 ) ↔ ∅ ∈ ( ( Fmla ‘ 𝑦 ) ∪ { 𝑥 ∣ ∃ 𝑢 ∈ ( Fmla ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( Fmla ‘ 𝑦 ) 𝑥 = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑢 ) } ) ) ) |
71 |
|
elun |
⊢ ( ∅ ∈ ( ( Fmla ‘ 𝑦 ) ∪ { 𝑥 ∣ ∃ 𝑢 ∈ ( Fmla ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( Fmla ‘ 𝑦 ) 𝑥 = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑢 ) } ) ↔ ( ∅ ∈ ( Fmla ‘ 𝑦 ) ∨ ∅ ∈ { 𝑥 ∣ ∃ 𝑢 ∈ ( Fmla ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( Fmla ‘ 𝑦 ) 𝑥 = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑢 ) } ) ) |
72 |
|
eqeq1 |
⊢ ( 𝑥 = ∅ → ( 𝑥 = ( 𝑢 ⊼𝑔 𝑣 ) ↔ ∅ = ( 𝑢 ⊼𝑔 𝑣 ) ) ) |
73 |
72
|
rexbidv |
⊢ ( 𝑥 = ∅ → ( ∃ 𝑣 ∈ ( Fmla ‘ 𝑦 ) 𝑥 = ( 𝑢 ⊼𝑔 𝑣 ) ↔ ∃ 𝑣 ∈ ( Fmla ‘ 𝑦 ) ∅ = ( 𝑢 ⊼𝑔 𝑣 ) ) ) |
74 |
|
eqeq1 |
⊢ ( 𝑥 = ∅ → ( 𝑥 = ∀𝑔 𝑖 𝑢 ↔ ∅ = ∀𝑔 𝑖 𝑢 ) ) |
75 |
74
|
rexbidv |
⊢ ( 𝑥 = ∅ → ( ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑢 ↔ ∃ 𝑖 ∈ ω ∅ = ∀𝑔 𝑖 𝑢 ) ) |
76 |
73 75
|
orbi12d |
⊢ ( 𝑥 = ∅ → ( ( ∃ 𝑣 ∈ ( Fmla ‘ 𝑦 ) 𝑥 = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑢 ) ↔ ( ∃ 𝑣 ∈ ( Fmla ‘ 𝑦 ) ∅ = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω ∅ = ∀𝑔 𝑖 𝑢 ) ) ) |
77 |
76
|
rexbidv |
⊢ ( 𝑥 = ∅ → ( ∃ 𝑢 ∈ ( Fmla ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( Fmla ‘ 𝑦 ) 𝑥 = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑢 ) ↔ ∃ 𝑢 ∈ ( Fmla ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( Fmla ‘ 𝑦 ) ∅ = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω ∅ = ∀𝑔 𝑖 𝑢 ) ) ) |
78 |
13 77
|
elab |
⊢ ( ∅ ∈ { 𝑥 ∣ ∃ 𝑢 ∈ ( Fmla ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( Fmla ‘ 𝑦 ) 𝑥 = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑢 ) } ↔ ∃ 𝑢 ∈ ( Fmla ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( Fmla ‘ 𝑦 ) ∅ = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω ∅ = ∀𝑔 𝑖 𝑢 ) ) |
79 |
78
|
orbi2i |
⊢ ( ( ∅ ∈ ( Fmla ‘ 𝑦 ) ∨ ∅ ∈ { 𝑥 ∣ ∃ 𝑢 ∈ ( Fmla ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( Fmla ‘ 𝑦 ) 𝑥 = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑢 ) } ) ↔ ( ∅ ∈ ( Fmla ‘ 𝑦 ) ∨ ∃ 𝑢 ∈ ( Fmla ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( Fmla ‘ 𝑦 ) ∅ = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω ∅ = ∀𝑔 𝑖 𝑢 ) ) ) |
80 |
71 79
|
bitri |
⊢ ( ∅ ∈ ( ( Fmla ‘ 𝑦 ) ∪ { 𝑥 ∣ ∃ 𝑢 ∈ ( Fmla ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( Fmla ‘ 𝑦 ) 𝑥 = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑢 ) } ) ↔ ( ∅ ∈ ( Fmla ‘ 𝑦 ) ∨ ∃ 𝑢 ∈ ( Fmla ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( Fmla ‘ 𝑦 ) ∅ = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω ∅ = ∀𝑔 𝑖 𝑢 ) ) ) |
81 |
70 80
|
bitrdi |
⊢ ( 𝑦 ∈ ω → ( ∅ ∈ ( Fmla ‘ suc 𝑦 ) ↔ ( ∅ ∈ ( Fmla ‘ 𝑦 ) ∨ ∃ 𝑢 ∈ ( Fmla ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( Fmla ‘ 𝑦 ) ∅ = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω ∅ = ∀𝑔 𝑖 𝑢 ) ) ) ) |
82 |
81
|
adantr |
⊢ ( ( 𝑦 ∈ ω ∧ ¬ ∅ ∈ ( Fmla ‘ 𝑦 ) ) → ( ∅ ∈ ( Fmla ‘ suc 𝑦 ) ↔ ( ∅ ∈ ( Fmla ‘ 𝑦 ) ∨ ∃ 𝑢 ∈ ( Fmla ‘ 𝑦 ) ( ∃ 𝑣 ∈ ( Fmla ‘ 𝑦 ) ∅ = ( 𝑢 ⊼𝑔 𝑣 ) ∨ ∃ 𝑖 ∈ ω ∅ = ∀𝑔 𝑖 𝑢 ) ) ) ) |
83 |
68 82
|
mtbird |
⊢ ( ( 𝑦 ∈ ω ∧ ¬ ∅ ∈ ( Fmla ‘ 𝑦 ) ) → ¬ ∅ ∈ ( Fmla ‘ suc 𝑦 ) ) |
84 |
83
|
ex |
⊢ ( 𝑦 ∈ ω → ( ¬ ∅ ∈ ( Fmla ‘ 𝑦 ) → ¬ ∅ ∈ ( Fmla ‘ suc 𝑦 ) ) ) |
85 |
3 6 9 12 35 84
|
finds |
⊢ ( 𝑁 ∈ ω → ¬ ∅ ∈ ( Fmla ‘ 𝑁 ) ) |
86 |
|
df-nel |
⊢ ( ∅ ∉ ( Fmla ‘ 𝑁 ) ↔ ¬ ∅ ∈ ( Fmla ‘ 𝑁 ) ) |
87 |
85 86
|
sylibr |
⊢ ( 𝑁 ∈ ω → ∅ ∉ ( Fmla ‘ 𝑁 ) ) |