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Theorem finxpreclem6 36580
Description: Lemma for ↑↑ recursion theorems. (Contributed by ML, 24-Oct-2020.)
Hypothesis
Ref Expression
finxpreclem5.1 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
Assertion
Ref Expression
finxpreclem6 ((𝑁 ∈ ω ∧ 1o𝑁) → (𝑈↑↑𝑁) ⊆ (V × 𝑈))
Distinct variable groups:   𝑥,𝑛,𝑁   𝑈,𝑛,𝑥
Allowed substitution hints:   𝐹(𝑥,𝑛)

Proof of Theorem finxpreclem6
Dummy variables 𝑚 𝑜 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2819 . . . . 5 (𝑛 = 𝑁 → (𝑛 ∈ ω ↔ 𝑁 ∈ ω))
2 eleq2 2820 . . . . 5 (𝑛 = 𝑁 → (1o𝑛 ↔ 1o𝑁))
31, 2anbi12d 629 . . . 4 (𝑛 = 𝑁 → ((𝑛 ∈ ω ∧ 1o𝑛) ↔ (𝑁 ∈ ω ∧ 1o𝑁)))
4 anass 467 . . . . . . . . 9 (((𝑛 ∈ ω ∧ 1o𝑛) ∧ ¬ 𝑦 ∈ (V × 𝑈)) ↔ (𝑛 ∈ ω ∧ (1o𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈))))
5 nfv 1915 . . . . . . . . . . . . . . 15 𝑥(𝑛 ∈ ω ∧ (1o𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈)))
6 finxpreclem5.1 . . . . . . . . . . . . . . . . . . . 20 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
7 nfmpo2 7492 . . . . . . . . . . . . . . . . . . . 20 𝑥(𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
86, 7nfcxfr 2899 . . . . . . . . . . . . . . . . . . 19 𝑥𝐹
9 nfcv 2901 . . . . . . . . . . . . . . . . . . 19 𝑥𝑛, 𝑦
108, 9nfrdg 8416 . . . . . . . . . . . . . . . . . 18 𝑥rec(𝐹, ⟨𝑛, 𝑦⟩)
11 nfcv 2901 . . . . . . . . . . . . . . . . . 18 𝑥𝑛
1210, 11nffv 6900 . . . . . . . . . . . . . . . . 17 𝑥(rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)
1312nfeq2 2918 . . . . . . . . . . . . . . . 16 𝑥∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)
1413nfn 1858 . . . . . . . . . . . . . . 15 𝑥 ¬ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)
155, 14nfim 1897 . . . . . . . . . . . . . 14 𝑥((𝑛 ∈ ω ∧ (1o𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈))) → ¬ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))
16 eleq1 2819 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑦 → (𝑥 ∈ (V × 𝑈) ↔ 𝑦 ∈ (V × 𝑈)))
1716notbid 317 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 → (¬ 𝑥 ∈ (V × 𝑈) ↔ ¬ 𝑦 ∈ (V × 𝑈)))
1817anbi2d 627 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → ((1o𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈)) ↔ (1o𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈))))
1918anbi2d 627 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → ((𝑛 ∈ ω ∧ (1o𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈))) ↔ (𝑛 ∈ ω ∧ (1o𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈)))))
20 opeq2 4873 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑦 → ⟨𝑛, 𝑥⟩ = ⟨𝑛, 𝑦⟩)
21 rdgeq2 8414 . . . . . . . . . . . . . . . . . . 19 (⟨𝑛, 𝑥⟩ = ⟨𝑛, 𝑦⟩ → rec(𝐹, ⟨𝑛, 𝑥⟩) = rec(𝐹, ⟨𝑛, 𝑦⟩))
2220, 21syl 17 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑦 → rec(𝐹, ⟨𝑛, 𝑥⟩) = rec(𝐹, ⟨𝑛, 𝑦⟩))
2322fveq1d 6892 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))
2423eqeq2d 2741 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → (∅ = (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) ↔ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)))
2524notbid 317 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (¬ ∅ = (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) ↔ ¬ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)))
2619, 25imbi12d 343 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (((𝑛 ∈ ω ∧ (1o𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈))) → ¬ ∅ = (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛)) ↔ ((𝑛 ∈ ω ∧ (1o𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈))) → ¬ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))))
27 anass 467 . . . . . . . . . . . . . . . . . . 19 (((𝑛 ∈ ω ∧ 1o𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) ↔ (𝑛 ∈ ω ∧ (1o𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈))))
28 vex 3476 . . . . . . . . . . . . . . . . . . . 20 𝑛 ∈ V
29 fveqeq2 6899 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 = ∅ → ((rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑚) = ⟨𝑛, 𝑥⟩ ↔ (rec(𝐹, ⟨𝑛, 𝑥⟩)‘∅) = ⟨𝑛, 𝑥⟩))
30 fveqeq2 6899 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 = 𝑜 → ((rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑚) = ⟨𝑛, 𝑥⟩ ↔ (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑜) = ⟨𝑛, 𝑥⟩))
31 fveqeq2 6899 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 = suc 𝑜 → ((rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑚) = ⟨𝑛, 𝑥⟩ ↔ (rec(𝐹, ⟨𝑛, 𝑥⟩)‘suc 𝑜) = ⟨𝑛, 𝑥⟩))
32 opex 5463 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑛, 𝑥⟩ ∈ V
3332rdg0 8423 . . . . . . . . . . . . . . . . . . . . . . . 24 (rec(𝐹, ⟨𝑛, 𝑥⟩)‘∅) = ⟨𝑛, 𝑥
3433a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑛 ∈ ω ∧ 1o𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘∅) = ⟨𝑛, 𝑥⟩)
35 nnon 7863 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑜 ∈ ω → 𝑜 ∈ On)
36 rdgsuc 8426 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑜 ∈ On → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘suc 𝑜) = (𝐹‘(rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑜)))
3735, 36syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑜 ∈ ω → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘suc 𝑜) = (𝐹‘(rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑜)))
38 fveq2 6890 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑜) = ⟨𝑛, 𝑥⟩ → (𝐹‘(rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑜)) = (𝐹‘⟨𝑛, 𝑥⟩))
3937, 38sylan9eq 2790 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑜 ∈ ω ∧ (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑜) = ⟨𝑛, 𝑥⟩) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘suc 𝑜) = (𝐹‘⟨𝑛, 𝑥⟩))
406finxpreclem5 36579 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑛 ∈ ω ∧ 1o𝑛) → (¬ 𝑥 ∈ (V × 𝑈) → (𝐹‘⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩))
4140imp 405 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑛 ∈ ω ∧ 1o𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (𝐹‘⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩)
4239, 41sylan9eq 2790 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑜 ∈ ω ∧ (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑜) = ⟨𝑛, 𝑥⟩) ∧ ((𝑛 ∈ ω ∧ 1o𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈))) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘suc 𝑜) = ⟨𝑛, 𝑥⟩)
4342expl 456 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑜 ∈ ω → (((rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑜) = ⟨𝑛, 𝑥⟩ ∧ ((𝑛 ∈ ω ∧ 1o𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈))) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘suc 𝑜) = ⟨𝑛, 𝑥⟩))
4443expcomd 415 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑜 ∈ ω → (((𝑛 ∈ ω ∧ 1o𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → ((rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑜) = ⟨𝑛, 𝑥⟩ → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘suc 𝑜) = ⟨𝑛, 𝑥⟩)))
4529, 30, 31, 34, 44finds2 7893 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 ∈ ω → (((𝑛 ∈ ω ∧ 1o𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑚) = ⟨𝑛, 𝑥⟩))
46 eleq1 2819 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = 𝑚 → (𝑛 ∈ ω ↔ 𝑚 ∈ ω))
47 fveqeq2 6899 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = 𝑚 → ((rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = ⟨𝑛, 𝑥⟩ ↔ (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑚) = ⟨𝑛, 𝑥⟩))
4847imbi2d 339 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = 𝑚 → ((((𝑛 ∈ ω ∧ 1o𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = ⟨𝑛, 𝑥⟩) ↔ (((𝑛 ∈ ω ∧ 1o𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑚) = ⟨𝑛, 𝑥⟩)))
4946, 48imbi12d 343 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝑚 → ((𝑛 ∈ ω → (((𝑛 ∈ ω ∧ 1o𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = ⟨𝑛, 𝑥⟩)) ↔ (𝑚 ∈ ω → (((𝑛 ∈ ω ∧ 1o𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑚) = ⟨𝑛, 𝑥⟩))))
5045, 49mpbiri 257 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑚 → (𝑛 ∈ ω → (((𝑛 ∈ ω ∧ 1o𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = ⟨𝑛, 𝑥⟩)))
5150equcoms 2021 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = 𝑛 → (𝑛 ∈ ω → (((𝑛 ∈ ω ∧ 1o𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = ⟨𝑛, 𝑥⟩)))
5228, 51vtocle 3542 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ω → (((𝑛 ∈ ω ∧ 1o𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = ⟨𝑛, 𝑥⟩))
5327, 52biimtrrid 242 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ω → ((𝑛 ∈ ω ∧ (1o𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈))) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = ⟨𝑛, 𝑥⟩))
5453anabsi5 665 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ ω ∧ (1o𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈))) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = ⟨𝑛, 𝑥⟩)
55 vex 3476 . . . . . . . . . . . . . . . . . . 19 𝑥 ∈ V
5628, 55opnzi 5473 . . . . . . . . . . . . . . . . . 18 𝑛, 𝑥⟩ ≠ ∅
5756a1i 11 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ ω ∧ (1o𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈))) → ⟨𝑛, 𝑥⟩ ≠ ∅)
5854, 57eqnetrd 3006 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ ω ∧ (1o𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈))) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) ≠ ∅)
5958necomd 2994 . . . . . . . . . . . . . . 15 ((𝑛 ∈ ω ∧ (1o𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈))) → ∅ ≠ (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛))
6059neneqd 2943 . . . . . . . . . . . . . 14 ((𝑛 ∈ ω ∧ (1o𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈))) → ¬ ∅ = (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛))
6115, 26, 60chvarfv 2231 . . . . . . . . . . . . 13 ((𝑛 ∈ ω ∧ (1o𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈))) → ¬ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))
6261intnand 487 . . . . . . . . . . . 12 ((𝑛 ∈ ω ∧ (1o𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈))) → ¬ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)))
6362adantl 480 . . . . . . . . . . 11 ((𝑛 = 𝑁 ∧ (𝑛 ∈ ω ∧ (1o𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈)))) → ¬ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)))
64 opeq1 4872 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝑁 → ⟨𝑛, 𝑦⟩ = ⟨𝑁, 𝑦⟩)
65 rdgeq2 8414 . . . . . . . . . . . . . . . . . . . 20 (⟨𝑛, 𝑦⟩ = ⟨𝑁, 𝑦⟩ → rec(𝐹, ⟨𝑛, 𝑦⟩) = rec(𝐹, ⟨𝑁, 𝑦⟩))
6664, 65syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑁 → rec(𝐹, ⟨𝑛, 𝑦⟩) = rec(𝐹, ⟨𝑁, 𝑦⟩))
67 id 22 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑁𝑛 = 𝑁)
6866, 67fveq12d 6897 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑁 → (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛) = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁))
6968eqeq2d 2741 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑁 → (∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛) ↔ ∅ = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁)))
701, 69anbi12d 629 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑁 → ((𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)) ↔ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁))))
7170abbidv 2799 . . . . . . . . . . . . . . 15 (𝑛 = 𝑁 → {𝑦 ∣ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))} = {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁))})
726dffinxpf 36569 . . . . . . . . . . . . . . 15 (𝑈↑↑𝑁) = {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁))}
7371, 72eqtr4di 2788 . . . . . . . . . . . . . 14 (𝑛 = 𝑁 → {𝑦 ∣ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))} = (𝑈↑↑𝑁))
7473eleq2d 2817 . . . . . . . . . . . . 13 (𝑛 = 𝑁 → (𝑦 ∈ {𝑦 ∣ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))} ↔ 𝑦 ∈ (𝑈↑↑𝑁)))
75 abid 2711 . . . . . . . . . . . . 13 (𝑦 ∈ {𝑦 ∣ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))} ↔ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)))
7674, 75bitr3di 285 . . . . . . . . . . . 12 (𝑛 = 𝑁 → (𝑦 ∈ (𝑈↑↑𝑁) ↔ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))))
7776adantr 479 . . . . . . . . . . 11 ((𝑛 = 𝑁 ∧ (𝑛 ∈ ω ∧ (1o𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈)))) → (𝑦 ∈ (𝑈↑↑𝑁) ↔ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))))
7863, 77mtbird 324 . . . . . . . . . 10 ((𝑛 = 𝑁 ∧ (𝑛 ∈ ω ∧ (1o𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈)))) → ¬ 𝑦 ∈ (𝑈↑↑𝑁))
7978ex 411 . . . . . . . . 9 (𝑛 = 𝑁 → ((𝑛 ∈ ω ∧ (1o𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈))) → ¬ 𝑦 ∈ (𝑈↑↑𝑁)))
804, 79biimtrid 241 . . . . . . . 8 (𝑛 = 𝑁 → (((𝑛 ∈ ω ∧ 1o𝑛) ∧ ¬ 𝑦 ∈ (V × 𝑈)) → ¬ 𝑦 ∈ (𝑈↑↑𝑁)))
8180expdimp 451 . . . . . . 7 ((𝑛 = 𝑁 ∧ (𝑛 ∈ ω ∧ 1o𝑛)) → (¬ 𝑦 ∈ (V × 𝑈) → ¬ 𝑦 ∈ (𝑈↑↑𝑁)))
8281con4d 115 . . . . . 6 ((𝑛 = 𝑁 ∧ (𝑛 ∈ ω ∧ 1o𝑛)) → (𝑦 ∈ (𝑈↑↑𝑁) → 𝑦 ∈ (V × 𝑈)))
8382ssrdv 3987 . . . . 5 ((𝑛 = 𝑁 ∧ (𝑛 ∈ ω ∧ 1o𝑛)) → (𝑈↑↑𝑁) ⊆ (V × 𝑈))
8483ex 411 . . . 4 (𝑛 = 𝑁 → ((𝑛 ∈ ω ∧ 1o𝑛) → (𝑈↑↑𝑁) ⊆ (V × 𝑈)))
853, 84sylbird 259 . . 3 (𝑛 = 𝑁 → ((𝑁 ∈ ω ∧ 1o𝑁) → (𝑈↑↑𝑁) ⊆ (V × 𝑈)))
8685vtocleg 3540 . 2 (𝑁 ∈ ω → ((𝑁 ∈ ω ∧ 1o𝑁) → (𝑈↑↑𝑁) ⊆ (V × 𝑈)))
8786anabsi5 665 1 ((𝑁 ∈ ω ∧ 1o𝑁) → (𝑈↑↑𝑁) ⊆ (V × 𝑈))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394   = wceq 1539  wcel 2104  {cab 2707  wne 2938  Vcvv 3472  wss 3947  c0 4321  ifcif 4527  cop 4633   cuni 4907   × cxp 5673  Oncon0 6363  suc csuc 6365  cfv 6542  cmpo 7413  ωcom 7857  1st c1st 7975  reccrdg 8411  1oc1o 8461  ↑↑cfinxp 36567
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-finxp 36568
This theorem is referenced by:  finxpsuclem  36581
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