| Step | Hyp | Ref
| Expression |
| 1 | | eleq1 2829 |
. . . . 5
⊢ (𝑛 = 𝑁 → (𝑛 ∈ ω ↔ 𝑁 ∈ ω)) |
| 2 | | eleq2 2830 |
. . . . 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 468 |
. . . . . . . . 9
⊢ (((𝑛 ∈ ω ∧
1o ∈ 𝑛)
∧ ¬ 𝑦 ∈ (V
× 𝑈)) ↔ (𝑛 ∈ ω ∧
(1o ∈ 𝑛
∧ ¬ 𝑦 ∈ (V
× 𝑈)))) |
| 5 | | nfv 1914 |
. . . . . . . . . . . . . . 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 7514 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑥(𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), 〈∪ 𝑛, (1st ‘𝑥)〉, 〈𝑛, 𝑥〉))) |
| 8 | 6, 7 | nfcxfr 2903 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑥𝐹 |
| 9 | | nfcv 2905 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑥〈𝑛, 𝑦〉 |
| 10 | 8, 9 | nfrdg 8454 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑥rec(𝐹, 〈𝑛, 𝑦〉) |
| 11 | | nfcv 2905 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑥𝑛 |
| 12 | 10, 11 | nffv 6916 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑥(rec(𝐹, 〈𝑛, 𝑦〉)‘𝑛) |
| 13 | 12 | nfeq2 2923 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑥∅ =
(rec(𝐹, 〈𝑛, 𝑦〉)‘𝑛) |
| 14 | 13 | nfn 1857 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑥 ¬
∅ = (rec(𝐹,
〈𝑛, 𝑦〉)‘𝑛) |
| 15 | 5, 14 | nfim 1896 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑥((𝑛 ∈ ω ∧
(1o ∈ 𝑛
∧ ¬ 𝑦 ∈ (V
× 𝑈))) → ¬
∅ = (rec(𝐹,
〈𝑛, 𝑦〉)‘𝑛)) |
| 16 | | eleq1 2829 |
. . . . . . . . . . . . . . . . . 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 4874 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 = 𝑦 → 〈𝑛, 𝑥〉 = 〈𝑛, 𝑦〉) |
| 21 | | rdgeq2 8452 |
. . . . . . . . . . . . . . . . . . 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 6908 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = 𝑦 → (rec(𝐹, 〈𝑛, 𝑥〉)‘𝑛) = (rec(𝐹, 〈𝑛, 𝑦〉)‘𝑛)) |
| 24 | 23 | eqeq2d 2748 |
. . . . . . . . . . . . . . . 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 344 |
. . . . . . . . . . . . . 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 468 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑛 ∈ ω ∧
1o ∈ 𝑛)
∧ ¬ 𝑥 ∈ (V
× 𝑈)) ↔ (𝑛 ∈ ω ∧
(1o ∈ 𝑛
∧ ¬ 𝑥 ∈ (V
× 𝑈)))) |
| 28 | | vex 3484 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 𝑛 ∈ V |
| 29 | | fveqeq2 6915 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑚 = ∅ → ((rec(𝐹, 〈𝑛, 𝑥〉)‘𝑚) = 〈𝑛, 𝑥〉 ↔ (rec(𝐹, 〈𝑛, 𝑥〉)‘∅) = 〈𝑛, 𝑥〉)) |
| 30 | | fveqeq2 6915 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑚 = 𝑜 → ((rec(𝐹, 〈𝑛, 𝑥〉)‘𝑚) = 〈𝑛, 𝑥〉 ↔ (rec(𝐹, 〈𝑛, 𝑥〉)‘𝑜) = 〈𝑛, 𝑥〉)) |
| 31 | | fveqeq2 6915 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑚 = suc 𝑜 → ((rec(𝐹, 〈𝑛, 𝑥〉)‘𝑚) = 〈𝑛, 𝑥〉 ↔ (rec(𝐹, 〈𝑛, 𝑥〉)‘suc 𝑜) = 〈𝑛, 𝑥〉)) |
| 32 | | opex 5469 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
〈𝑛, 𝑥〉 ∈ V |
| 33 | 32 | rdg0 8461 |
. . . . . . . . . . . . . . . . . . . . . . . 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 7893 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑜 ∈ ω → 𝑜 ∈ On) |
| 36 | | rdgsuc 8464 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6906 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢
((rec(𝐹, 〈𝑛, 𝑥〉)‘𝑜) = 〈𝑛, 𝑥〉 → (𝐹‘(rec(𝐹, 〈𝑛, 𝑥〉)‘𝑜)) = (𝐹‘〈𝑛, 𝑥〉)) |
| 39 | 37, 38 | sylan9eq 2797 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑜 ∈ ω ∧ (rec(𝐹, 〈𝑛, 𝑥〉)‘𝑜) = 〈𝑛, 𝑥〉) → (rec(𝐹, 〈𝑛, 𝑥〉)‘suc 𝑜) = (𝐹‘〈𝑛, 𝑥〉)) |
| 40 | 6 | finxpreclem5 37396 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑛 ∈ ω ∧
1o ∈ 𝑛)
→ (¬ 𝑥 ∈ (V
× 𝑈) → (𝐹‘〈𝑛, 𝑥〉) = 〈𝑛, 𝑥〉)) |
| 41 | 40 | imp 406 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝑛 ∈ ω ∧
1o ∈ 𝑛)
∧ ¬ 𝑥 ∈ (V
× 𝑈)) → (𝐹‘〈𝑛, 𝑥〉) = 〈𝑛, 𝑥〉) |
| 42 | 39, 41 | sylan9eq 2797 |
. . . . . . . . . . . . . . . . . . . . . . . . 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 457 |
. . . . . . . . . . . . . . . . . . . . . . . 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 416 |
. . . . . . . . . . . . . . . . . . . . . . 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 7920 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑚 ∈ ω → (((𝑛 ∈ ω ∧
1o ∈ 𝑛)
∧ ¬ 𝑥 ∈ (V
× 𝑈)) →
(rec(𝐹, 〈𝑛, 𝑥〉)‘𝑚) = 〈𝑛, 𝑥〉)) |
| 46 | | eleq1 2829 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑛 = 𝑚 → (𝑛 ∈ ω ↔ 𝑚 ∈ ω)) |
| 47 | | fveqeq2 6915 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑛 = 𝑚 → ((rec(𝐹, 〈𝑛, 𝑥〉)‘𝑛) = 〈𝑛, 𝑥〉 ↔ (rec(𝐹, 〈𝑛, 𝑥〉)‘𝑚) = 〈𝑛, 𝑥〉)) |
| 48 | 47 | imbi2d 340 |
. . . . . . . . . . . . . . . . . . . . . . 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 344 |
. . . . . . . . . . . . . . . . . . . . . 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 2019 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑚 = 𝑛 → (𝑛 ∈ ω → (((𝑛 ∈ ω ∧ 1o ∈
𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (rec(𝐹, 〈𝑛, 𝑥〉)‘𝑛) = 〈𝑛, 𝑥〉))) |
| 52 | 28, 51 | vtocle 3555 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 ∈ ω → (((𝑛 ∈ ω ∧
1o ∈ 𝑛)
∧ ¬ 𝑥 ∈ (V
× 𝑈)) →
(rec(𝐹, 〈𝑛, 𝑥〉)‘𝑛) = 〈𝑛, 𝑥〉)) |
| 53 | 27, 52 | biimtrrid 243 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ ω → ((𝑛 ∈ ω ∧
(1o ∈ 𝑛
∧ ¬ 𝑥 ∈ (V
× 𝑈))) →
(rec(𝐹, 〈𝑛, 𝑥〉)‘𝑛) = 〈𝑛, 𝑥〉)) |
| 54 | 53 | anabsi5 669 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑛 ∈ ω ∧
(1o ∈ 𝑛
∧ ¬ 𝑥 ∈ (V
× 𝑈))) →
(rec(𝐹, 〈𝑛, 𝑥〉)‘𝑛) = 〈𝑛, 𝑥〉) |
| 55 | | vex 3484 |
. . . . . . . . . . . . . . . . . . 19
⊢ 𝑥 ∈ V |
| 56 | 28, 55 | opnzi 5479 |
. . . . . . . . . . . . . . . . . 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 2240 |
. . . . . . . . . . . . 13
⊢ ((𝑛 ∈ ω ∧
(1o ∈ 𝑛
∧ ¬ 𝑦 ∈ (V
× 𝑈))) → ¬
∅ = (rec(𝐹,
〈𝑛, 𝑦〉)‘𝑛)) |
| 62 | 61 | intnand 488 |
. . . . . . . . . . . 12
⊢ ((𝑛 ∈ ω ∧
(1o ∈ 𝑛
∧ ¬ 𝑦 ∈ (V
× 𝑈))) → ¬
(𝑛 ∈ ω ∧
∅ = (rec(𝐹,
〈𝑛, 𝑦〉)‘𝑛))) |
| 63 | 62 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝑛 = 𝑁 ∧ (𝑛 ∈ ω ∧ (1o ∈
𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈)))) → ¬ (𝑛 ∈ ω ∧ ∅ =
(rec(𝐹, 〈𝑛, 𝑦〉)‘𝑛))) |
| 64 | | opeq1 4873 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 = 𝑁 → 〈𝑛, 𝑦〉 = 〈𝑁, 𝑦〉) |
| 65 | | rdgeq2 8452 |
. . . . . . . . . . . . . . . . . . . 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 6913 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 = 𝑁 → (rec(𝐹, 〈𝑛, 𝑦〉)‘𝑛) = (rec(𝐹, 〈𝑁, 𝑦〉)‘𝑁)) |
| 69 | 68 | eqeq2d 2748 |
. . . . . . . . . . . . . . . . 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 2808 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 𝑁 → {𝑦 ∣ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, 〈𝑛, 𝑦〉)‘𝑛))} = {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, 〈𝑁, 𝑦〉)‘𝑁))}) |
| 72 | 6 | dffinxpf 37386 |
. . . . . . . . . . . . . . 15
⊢ (𝑈↑↑𝑁) = {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, 〈𝑁, 𝑦〉)‘𝑁))} |
| 73 | 71, 72 | eqtr4di 2795 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = 𝑁 → {𝑦 ∣ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, 〈𝑛, 𝑦〉)‘𝑛))} = (𝑈↑↑𝑁)) |
| 74 | 73 | eleq2d 2827 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 𝑁 → (𝑦 ∈ {𝑦 ∣ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, 〈𝑛, 𝑦〉)‘𝑛))} ↔ 𝑦 ∈ (𝑈↑↑𝑁))) |
| 75 | | abid 2718 |
. . . . . . . . . . . . 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 480 |
. . . . . . . . . . 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 412 |
. . . . . . . . 9
⊢ (𝑛 = 𝑁 → ((𝑛 ∈ ω ∧ (1o ∈
𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈))) → ¬ 𝑦 ∈ (𝑈↑↑𝑁))) |
| 80 | 4, 79 | biimtrid 242 |
. . . . . . . 8
⊢ (𝑛 = 𝑁 → (((𝑛 ∈ ω ∧ 1o ∈
𝑛) ∧ ¬ 𝑦 ∈ (V × 𝑈)) → ¬ 𝑦 ∈ (𝑈↑↑𝑁))) |
| 81 | 80 | expdimp 452 |
. . . . . . 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 3989 |
. . . . 5
⊢ ((𝑛 = 𝑁 ∧ (𝑛 ∈ ω ∧ 1o ∈
𝑛)) → (𝑈↑↑𝑁) ⊆ (V × 𝑈)) |
| 84 | 83 | ex 412 |
. . . 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 3553 |
. 2
⊢ (𝑁 ∈ ω → ((𝑁 ∈ ω ∧
1o ∈ 𝑁)
→ (𝑈↑↑𝑁) ⊆ (V × 𝑈))) |
| 87 | 86 | anabsi5 669 |
1
⊢ ((𝑁 ∈ ω ∧
1o ∈ 𝑁)
→ (𝑈↑↑𝑁) ⊆ (V × 𝑈)) |