Step | Hyp | Ref
| Expression |
1 | | eleq1 2822 |
. . . . 5
⊢ (𝑛 = 𝑁 → (𝑛 ∈ ω ↔ 𝑁 ∈ ω)) |
2 | | eleq2 2823 |
. . . . 5
⊢ (𝑛 = 𝑁 → (1o ∈ 𝑛 ↔ 1o ∈
𝑁)) |
3 | 1, 2 | anbi12d 632 |
. . . 4
⊢ (𝑛 = 𝑁 → ((𝑛 ∈ ω ∧ 1o ∈
𝑛) ↔ (𝑁 ∈ ω ∧
1o ∈ 𝑁))) |
4 | | anass 470 |
. . . . . . . . 9
⊢ (((𝑛 ∈ ω ∧
1o ∈ 𝑛)
∧ ¬ 𝑦 ∈ (V
× 𝑈)) ↔ (𝑛 ∈ ω ∧
(1o ∈ 𝑛
∧ ¬ 𝑦 ∈ (V
× 𝑈)))) |
5 | | nfv 1918 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑥(𝑛 ∈ ω ∧
(1o ∈ 𝑛
∧ ¬ 𝑦 ∈ (V
× 𝑈))) |
6 | | finxpreclem5.1 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))) |
7 | | nfmpo2 7442 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑥(𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))) |
8 | 6, 7 | nfcxfr 2902 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑥𝐹 |
9 | | nfcv 2904 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑥⟨𝑛, 𝑦⟩ |
10 | 8, 9 | nfrdg 8364 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑥rec(𝐹, ⟨𝑛, 𝑦⟩) |
11 | | nfcv 2904 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑥𝑛 |
12 | 10, 11 | nffv 6856 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑥(rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛) |
13 | 12 | nfeq2 2921 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑥∅ =
(rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛) |
14 | 13 | nfn 1861 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑥 ¬
∅ = (rec(𝐹,
⟨𝑛, 𝑦⟩)‘𝑛) |
15 | 5, 14 | nfim 1900 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑥((𝑛 ∈ ω ∧
(1o ∈ 𝑛
∧ ¬ 𝑦 ∈ (V
× 𝑈))) → ¬
∅ = (rec(𝐹,
⟨𝑛, 𝑦⟩)‘𝑛)) |
16 | | eleq1 2822 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = 𝑦 → (𝑥 ∈ (V × 𝑈) ↔ 𝑦 ∈ (V × 𝑈))) |
17 | 16 | notbid 318 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ (V × 𝑈) ↔ ¬ 𝑦 ∈ (V × 𝑈))) |
18 | 17 | anbi2d 630 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = 𝑦 → ((1o ∈ 𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈)) ↔ (1o ∈ 𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈)))) |
19 | 18 | anbi2d 630 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝑦 → ((𝑛 ∈ ω ∧ (1o ∈
𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈))) ↔ (𝑛 ∈ ω ∧ (1o ∈
𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈))))) |
20 | | opeq2 4835 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 = 𝑦 → ⟨𝑛, 𝑥⟩ = ⟨𝑛, 𝑦⟩) |
21 | | rdgeq2 8362 |
. . . . . . . . . . . . . . . . . . 19
⊢
(⟨𝑛, 𝑥⟩ = ⟨𝑛, 𝑦⟩ → rec(𝐹, ⟨𝑛, 𝑥⟩) = rec(𝐹, ⟨𝑛, 𝑦⟩)) |
22 | 20, 21 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = 𝑦 → rec(𝐹, ⟨𝑛, 𝑥⟩) = rec(𝐹, ⟨𝑛, 𝑦⟩)) |
23 | 22 | fveq1d 6848 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = 𝑦 → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)) |
24 | 23 | eqeq2d 2744 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = 𝑦 → (∅ = (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) ↔ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))) |
25 | 24 | notbid 318 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝑦 → (¬ ∅ = (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) ↔ ¬ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))) |
26 | 19, 25 | imbi12d 345 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = 𝑦 → (((𝑛 ∈ ω ∧ (1o ∈
𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈))) → ¬ ∅ =
(rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛)) ↔ ((𝑛 ∈ ω ∧ (1o ∈
𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈))) → ¬ ∅ =
(rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)))) |
27 | | anass 470 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑛 ∈ ω ∧
1o ∈ 𝑛)
∧ ¬ 𝑥 ∈ (V
× 𝑈)) ↔ (𝑛 ∈ ω ∧
(1o ∈ 𝑛
∧ ¬ 𝑥 ∈ (V
× 𝑈)))) |
28 | | vex 3451 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 𝑛 ∈ V |
29 | | fveqeq2 6855 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑚 = ∅ → ((rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑚) = ⟨𝑛, 𝑥⟩ ↔ (rec(𝐹, ⟨𝑛, 𝑥⟩)‘∅) = ⟨𝑛, 𝑥⟩)) |
30 | | fveqeq2 6855 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑚 = 𝑜 → ((rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑚) = ⟨𝑛, 𝑥⟩ ↔ (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑜) = ⟨𝑛, 𝑥⟩)) |
31 | | fveqeq2 6855 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑚 = suc 𝑜 → ((rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑚) = ⟨𝑛, 𝑥⟩ ↔ (rec(𝐹, ⟨𝑛, 𝑥⟩)‘suc 𝑜) = ⟨𝑛, 𝑥⟩)) |
32 | | opex 5425 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
⟨𝑛, 𝑥⟩ ∈ V |
33 | 32 | rdg0 8371 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(rec(𝐹, ⟨𝑛, 𝑥⟩)‘∅) = ⟨𝑛, 𝑥⟩ |
34 | 33 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑛 ∈ ω ∧
1o ∈ 𝑛)
∧ ¬ 𝑥 ∈ (V
× 𝑈)) →
(rec(𝐹, ⟨𝑛, 𝑥⟩)‘∅) = ⟨𝑛, 𝑥⟩) |
35 | | nnon 7812 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑜 ∈ ω → 𝑜 ∈ On) |
36 | | rdgsuc 8374 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑜 ∈ On → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘suc 𝑜) = (𝐹‘(rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑜))) |
37 | 35, 36 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑜 ∈ ω →
(rec(𝐹, ⟨𝑛, 𝑥⟩)‘suc 𝑜) = (𝐹‘(rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑜))) |
38 | | fveq2 6846 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢
((rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑜) = ⟨𝑛, 𝑥⟩ → (𝐹‘(rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑜)) = (𝐹‘⟨𝑛, 𝑥⟩)) |
39 | 37, 38 | sylan9eq 2793 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑜 ∈ ω ∧ (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑜) = ⟨𝑛, 𝑥⟩) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘suc 𝑜) = (𝐹‘⟨𝑛, 𝑥⟩)) |
40 | 6 | finxpreclem5 35916 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑛 ∈ ω ∧
1o ∈ 𝑛)
→ (¬ 𝑥 ∈ (V
× 𝑈) → (𝐹‘⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩)) |
41 | 40 | imp 408 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝑛 ∈ ω ∧
1o ∈ 𝑛)
∧ ¬ 𝑥 ∈ (V
× 𝑈)) → (𝐹‘⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩) |
42 | 39, 41 | sylan9eq 2793 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝑜 ∈ ω ∧ (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑜) = ⟨𝑛, 𝑥⟩) ∧ ((𝑛 ∈ ω ∧ 1o ∈
𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈))) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘suc 𝑜) = ⟨𝑛, 𝑥⟩) |
43 | 42 | expl 459 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑜 ∈ ω →
(((rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑜) = ⟨𝑛, 𝑥⟩ ∧ ((𝑛 ∈ ω ∧ 1o ∈
𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈))) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘suc 𝑜) = ⟨𝑛, 𝑥⟩)) |
44 | 43 | expcomd 418 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑜 ∈ ω → (((𝑛 ∈ ω ∧
1o ∈ 𝑛)
∧ ¬ 𝑥 ∈ (V
× 𝑈)) →
((rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑜) = ⟨𝑛, 𝑥⟩ → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘suc 𝑜) = ⟨𝑛, 𝑥⟩))) |
45 | 29, 30, 31, 34, 44 | finds2 7841 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑚 ∈ ω → (((𝑛 ∈ ω ∧
1o ∈ 𝑛)
∧ ¬ 𝑥 ∈ (V
× 𝑈)) →
(rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑚) = ⟨𝑛, 𝑥⟩)) |
46 | | eleq1 2822 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑛 = 𝑚 → (𝑛 ∈ ω ↔ 𝑚 ∈ ω)) |
47 | | fveqeq2 6855 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑛 = 𝑚 → ((rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = ⟨𝑛, 𝑥⟩ ↔ (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑚) = ⟨𝑛, 𝑥⟩)) |
48 | 47 | imbi2d 341 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑛 = 𝑚 → ((((𝑛 ∈ ω ∧ 1o ∈
𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = ⟨𝑛, 𝑥⟩) ↔ (((𝑛 ∈ ω ∧ 1o ∈
𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑚) = ⟨𝑛, 𝑥⟩))) |
49 | 46, 48 | imbi12d 345 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑛 = 𝑚 → ((𝑛 ∈ ω → (((𝑛 ∈ ω ∧ 1o ∈
𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = ⟨𝑛, 𝑥⟩)) ↔ (𝑚 ∈ ω → (((𝑛 ∈ ω ∧ 1o ∈
𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑚) = ⟨𝑛, 𝑥⟩)))) |
50 | 45, 49 | mpbiri 258 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑛 = 𝑚 → (𝑛 ∈ ω → (((𝑛 ∈ ω ∧ 1o ∈
𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = ⟨𝑛, 𝑥⟩))) |
51 | 50 | equcoms 2024 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑚 = 𝑛 → (𝑛 ∈ ω → (((𝑛 ∈ ω ∧ 1o ∈
𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = ⟨𝑛, 𝑥⟩))) |
52 | 28, 51 | vtocle 3546 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 ∈ ω → (((𝑛 ∈ ω ∧
1o ∈ 𝑛)
∧ ¬ 𝑥 ∈ (V
× 𝑈)) →
(rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = ⟨𝑛, 𝑥⟩)) |
53 | 27, 52 | biimtrrid 242 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ ω → ((𝑛 ∈ ω ∧
(1o ∈ 𝑛
∧ ¬ 𝑥 ∈ (V
× 𝑈))) →
(rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = ⟨𝑛, 𝑥⟩)) |
54 | 53 | anabsi5 668 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑛 ∈ ω ∧
(1o ∈ 𝑛
∧ ¬ 𝑥 ∈ (V
× 𝑈))) →
(rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = ⟨𝑛, 𝑥⟩) |
55 | | vex 3451 |
. . . . . . . . . . . . . . . . . . 19
⊢ 𝑥 ∈ V |
56 | 28, 55 | opnzi 5435 |
. . . . . . . . . . . . . . . . . 18
⊢
⟨𝑛, 𝑥⟩ ≠
∅ |
57 | 56 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑛 ∈ ω ∧
(1o ∈ 𝑛
∧ ¬ 𝑥 ∈ (V
× 𝑈))) →
⟨𝑛, 𝑥⟩ ≠ ∅) |
58 | 54, 57 | eqnetrd 3008 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑛 ∈ ω ∧
(1o ∈ 𝑛
∧ ¬ 𝑥 ∈ (V
× 𝑈))) →
(rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) ≠ ∅) |
59 | 58 | necomd 2996 |
. . . . . . . . . . . . . . 15
⊢ ((𝑛 ∈ ω ∧
(1o ∈ 𝑛
∧ ¬ 𝑥 ∈ (V
× 𝑈))) → ∅
≠ (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛)) |
60 | 59 | neneqd 2945 |
. . . . . . . . . . . . . 14
⊢ ((𝑛 ∈ ω ∧
(1o ∈ 𝑛
∧ ¬ 𝑥 ∈ (V
× 𝑈))) → ¬
∅ = (rec(𝐹,
⟨𝑛, 𝑥⟩)‘𝑛)) |
61 | 15, 26, 60 | chvarfv 2234 |
. . . . . . . . . . . . 13
⊢ ((𝑛 ∈ ω ∧
(1o ∈ 𝑛
∧ ¬ 𝑦 ∈ (V
× 𝑈))) → ¬
∅ = (rec(𝐹,
⟨𝑛, 𝑦⟩)‘𝑛)) |
62 | 61 | intnand 490 |
. . . . . . . . . . . 12
⊢ ((𝑛 ∈ ω ∧
(1o ∈ 𝑛
∧ ¬ 𝑦 ∈ (V
× 𝑈))) → ¬
(𝑛 ∈ ω ∧
∅ = (rec(𝐹,
⟨𝑛, 𝑦⟩)‘𝑛))) |
63 | 62 | adantl 483 |
. . . . . . . . . . 11
⊢ ((𝑛 = 𝑁 ∧ (𝑛 ∈ ω ∧ (1o ∈
𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈)))) → ¬ (𝑛 ∈ ω ∧ ∅ =
(rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))) |
64 | | opeq1 4834 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 = 𝑁 → ⟨𝑛, 𝑦⟩ = ⟨𝑁, 𝑦⟩) |
65 | | rdgeq2 8362 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(⟨𝑛, 𝑦⟩ = ⟨𝑁, 𝑦⟩ → rec(𝐹, ⟨𝑛, 𝑦⟩) = rec(𝐹, ⟨𝑁, 𝑦⟩)) |
66 | 64, 65 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 = 𝑁 → rec(𝐹, ⟨𝑛, 𝑦⟩) = rec(𝐹, ⟨𝑁, 𝑦⟩)) |
67 | | id 22 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 = 𝑁 → 𝑛 = 𝑁) |
68 | 66, 67 | fveq12d 6853 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 = 𝑁 → (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛) = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁)) |
69 | 68 | eqeq2d 2744 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 = 𝑁 → (∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛) ↔ ∅ = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁))) |
70 | 1, 69 | anbi12d 632 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = 𝑁 → ((𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)) ↔ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁)))) |
71 | 70 | abbidv 2802 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 𝑁 → {𝑦 ∣ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))} = {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁))}) |
72 | 6 | dffinxpf 35906 |
. . . . . . . . . . . . . . 15
⊢ (𝑈↑↑𝑁) = {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁))} |
73 | 71, 72 | eqtr4di 2791 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = 𝑁 → {𝑦 ∣ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))} = (𝑈↑↑𝑁)) |
74 | 73 | eleq2d 2820 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 𝑁 → (𝑦 ∈ {𝑦 ∣ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))} ↔ 𝑦 ∈ (𝑈↑↑𝑁))) |
75 | | abid 2714 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ {𝑦 ∣ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))} ↔ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))) |
76 | 74, 75 | bitr3di 286 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝑁 → (𝑦 ∈ (𝑈↑↑𝑁) ↔ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)))) |
77 | 76 | adantr 482 |
. . . . . . . . . . 11
⊢ ((𝑛 = 𝑁 ∧ (𝑛 ∈ ω ∧ (1o ∈
𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈)))) → (𝑦 ∈ (𝑈↑↑𝑁) ↔ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)))) |
78 | 63, 77 | mtbird 325 |
. . . . . . . . . 10
⊢ ((𝑛 = 𝑁 ∧ (𝑛 ∈ ω ∧ (1o ∈
𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈)))) → ¬ 𝑦 ∈ (𝑈↑↑𝑁)) |
79 | 78 | ex 414 |
. . . . . . . . 9
⊢ (𝑛 = 𝑁 → ((𝑛 ∈ ω ∧ (1o ∈
𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈))) → ¬ 𝑦 ∈ (𝑈↑↑𝑁))) |
80 | 4, 79 | biimtrid 241 |
. . . . . . . 8
⊢ (𝑛 = 𝑁 → (((𝑛 ∈ ω ∧ 1o ∈
𝑛) ∧ ¬ 𝑦 ∈ (V × 𝑈)) → ¬ 𝑦 ∈ (𝑈↑↑𝑁))) |
81 | 80 | expdimp 454 |
. . . . . . 7
⊢ ((𝑛 = 𝑁 ∧ (𝑛 ∈ ω ∧ 1o ∈
𝑛)) → (¬ 𝑦 ∈ (V × 𝑈) → ¬ 𝑦 ∈ (𝑈↑↑𝑁))) |
82 | 81 | con4d 115 |
. . . . . 6
⊢ ((𝑛 = 𝑁 ∧ (𝑛 ∈ ω ∧ 1o ∈
𝑛)) → (𝑦 ∈ (𝑈↑↑𝑁) → 𝑦 ∈ (V × 𝑈))) |
83 | 82 | ssrdv 3954 |
. . . . 5
⊢ ((𝑛 = 𝑁 ∧ (𝑛 ∈ ω ∧ 1o ∈
𝑛)) → (𝑈↑↑𝑁) ⊆ (V × 𝑈)) |
84 | 83 | ex 414 |
. . . 4
⊢ (𝑛 = 𝑁 → ((𝑛 ∈ ω ∧ 1o ∈
𝑛) → (𝑈↑↑𝑁) ⊆ (V × 𝑈))) |
85 | 3, 84 | sylbird 260 |
. . 3
⊢ (𝑛 = 𝑁 → ((𝑁 ∈ ω ∧ 1o ∈
𝑁) → (𝑈↑↑𝑁) ⊆ (V × 𝑈))) |
86 | 85 | vtocleg 3516 |
. 2
⊢ (𝑁 ∈ ω → ((𝑁 ∈ ω ∧
1o ∈ 𝑁)
→ (𝑈↑↑𝑁) ⊆ (V × 𝑈))) |
87 | 86 | anabsi5 668 |
1
⊢ ((𝑁 ∈ ω ∧
1o ∈ 𝑁)
→ (𝑈↑↑𝑁) ⊆ (V × 𝑈)) |