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Theorem finxpreclem6 34814
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 2880 . . . . 5 (𝑛 = 𝑁 → (𝑛 ∈ ω ↔ 𝑁 ∈ ω))
2 eleq2 2881 . . . . 5 (𝑛 = 𝑁 → (1o𝑛 ↔ 1o𝑁))
31, 2anbi12d 633 . . . 4 (𝑛 = 𝑁 → ((𝑛 ∈ ω ∧ 1o𝑛) ↔ (𝑁 ∈ ω ∧ 1o𝑁)))
4 anass 472 . . . . . . . . 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 7218 . . . . . . . . . . . . . . . . . . . 20 𝑥(𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
86, 7nfcxfr 2956 . . . . . . . . . . . . . . . . . . 19 𝑥𝐹
9 nfcv 2958 . . . . . . . . . . . . . . . . . . 19 𝑥𝑛, 𝑦
108, 9nfrdg 8037 . . . . . . . . . . . . . . . . . 18 𝑥rec(𝐹, ⟨𝑛, 𝑦⟩)
11 nfcv 2958 . . . . . . . . . . . . . . . . . 18 𝑥𝑛
1210, 11nffv 6659 . . . . . . . . . . . . . . . . 17 𝑥(rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)
1312nfeq2 2975 . . . . . . . . . . . . . . . 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 2880 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑦 → (𝑥 ∈ (V × 𝑈) ↔ 𝑦 ∈ (V × 𝑈)))
1716notbid 321 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 → (¬ 𝑥 ∈ (V × 𝑈) ↔ ¬ 𝑦 ∈ (V × 𝑈)))
1817anbi2d 631 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → ((1o𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈)) ↔ (1o𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈))))
1918anbi2d 631 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → ((𝑛 ∈ ω ∧ (1o𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈))) ↔ (𝑛 ∈ ω ∧ (1o𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈)))))
20 opeq2 4768 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑦 → ⟨𝑛, 𝑥⟩ = ⟨𝑛, 𝑦⟩)
21 rdgeq2 8035 . . . . . . . . . . . . . . . . . . 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 6651 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))
2423eqeq2d 2812 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → (∅ = (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) ↔ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)))
2524notbid 321 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (¬ ∅ = (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) ↔ ¬ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)))
2619, 25imbi12d 348 . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . . . . . 19 (((𝑛 ∈ ω ∧ 1o𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) ↔ (𝑛 ∈ ω ∧ (1o𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈))))
28 vex 3447 . . . . . . . . . . . . . . . . . . . 20 𝑛 ∈ V
29 fveqeq2 6658 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 = ∅ → ((rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑚) = ⟨𝑛, 𝑥⟩ ↔ (rec(𝐹, ⟨𝑛, 𝑥⟩)‘∅) = ⟨𝑛, 𝑥⟩))
30 fveqeq2 6658 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 = 𝑜 → ((rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑚) = ⟨𝑛, 𝑥⟩ ↔ (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑜) = ⟨𝑛, 𝑥⟩))
31 fveqeq2 6658 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 = suc 𝑜 → ((rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑚) = ⟨𝑛, 𝑥⟩ ↔ (rec(𝐹, ⟨𝑛, 𝑥⟩)‘suc 𝑜) = ⟨𝑛, 𝑥⟩))
32 opex 5324 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑛, 𝑥⟩ ∈ V
3332rdg0 8044 . . . . . . . . . . . . . . . . . . . . . . . 24 (rec(𝐹, ⟨𝑛, 𝑥⟩)‘∅) = ⟨𝑛, 𝑥
3433a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑛 ∈ ω ∧ 1o𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘∅) = ⟨𝑛, 𝑥⟩)
35 nnon 7570 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑜 ∈ ω → 𝑜 ∈ On)
36 rdgsuc 8047 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6649 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑜) = ⟨𝑛, 𝑥⟩ → (𝐹‘(rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑜)) = (𝐹‘⟨𝑛, 𝑥⟩))
3937, 38sylan9eq 2856 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑜 ∈ ω ∧ (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑜) = ⟨𝑛, 𝑥⟩) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘suc 𝑜) = (𝐹‘⟨𝑛, 𝑥⟩))
406finxpreclem5 34813 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑛 ∈ ω ∧ 1o𝑛) → (¬ 𝑥 ∈ (V × 𝑈) → (𝐹‘⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩))
4140imp 410 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑛 ∈ ω ∧ 1o𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (𝐹‘⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩)
4239, 41sylan9eq 2856 . . . . . . . . . . . . . . . . . . . . . . . . 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 461 . . . . . . . . . . . . . . . . . . . . . . . 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 420 . . . . . . . . . . . . . . . . . . . . . . 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 7595 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 ∈ ω → (((𝑛 ∈ ω ∧ 1o𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑚) = ⟨𝑛, 𝑥⟩))
46 eleq1 2880 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = 𝑚 → (𝑛 ∈ ω ↔ 𝑚 ∈ ω))
47 fveqeq2 6658 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = 𝑚 → ((rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = ⟨𝑛, 𝑥⟩ ↔ (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑚) = ⟨𝑛, 𝑥⟩))
4847imbi2d 344 . . . . . . . . . . . . . . . . . . . . . . 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 348 . . . . . . . . . . . . . . . . . . . . . 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 261 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑚 → (𝑛 ∈ ω → (((𝑛 ∈ ω ∧ 1o𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = ⟨𝑛, 𝑥⟩)))
5150equcoms 2027 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = 𝑛 → (𝑛 ∈ ω → (((𝑛 ∈ ω ∧ 1o𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = ⟨𝑛, 𝑥⟩)))
5228, 51vtocle 3535 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ω → (((𝑛 ∈ ω ∧ 1o𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = ⟨𝑛, 𝑥⟩))
5327, 52syl5bir 246 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ω → ((𝑛 ∈ ω ∧ (1o𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈))) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = ⟨𝑛, 𝑥⟩))
5453anabsi5 668 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ ω ∧ (1o𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈))) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = ⟨𝑛, 𝑥⟩)
55 vex 3447 . . . . . . . . . . . . . . . . . . 19 𝑥 ∈ V
5628, 55opnzi 5334 . . . . . . . . . . . . . . . . . 18 𝑛, 𝑥⟩ ≠ ∅
5756a1i 11 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ ω ∧ (1o𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈))) → ⟨𝑛, 𝑥⟩ ≠ ∅)
5854, 57eqnetrd 3057 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ ω ∧ (1o𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈))) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) ≠ ∅)
5958necomd 3045 . . . . . . . . . . . . . . 15 ((𝑛 ∈ ω ∧ (1o𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈))) → ∅ ≠ (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛))
6059neneqd 2995 . . . . . . . . . . . . . 14 ((𝑛 ∈ ω ∧ (1o𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈))) → ¬ ∅ = (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛))
6115, 26, 60chvarfv 2241 . . . . . . . . . . . . 13 ((𝑛 ∈ ω ∧ (1o𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈))) → ¬ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))
6261intnand 492 . . . . . . . . . . . 12 ((𝑛 ∈ ω ∧ (1o𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈))) → ¬ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)))
6362adantl 485 . . . . . . . . . . 11 ((𝑛 = 𝑁 ∧ (𝑛 ∈ ω ∧ (1o𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈)))) → ¬ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)))
64 opeq1 4766 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝑁 → ⟨𝑛, 𝑦⟩ = ⟨𝑁, 𝑦⟩)
65 rdgeq2 8035 . . . . . . . . . . . . . . . . . . . 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 6656 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑁 → (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛) = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁))
6968eqeq2d 2812 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑁 → (∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛) ↔ ∅ = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁)))
701, 69anbi12d 633 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑁 → ((𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)) ↔ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁))))
7170abbidv 2865 . . . . . . . . . . . . . . 15 (𝑛 = 𝑁 → {𝑦 ∣ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))} = {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁))})
726dffinxpf 34803 . . . . . . . . . . . . . . 15 (𝑈↑↑𝑁) = {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁))}
7371, 72eqtr4di 2854 . . . . . . . . . . . . . 14 (𝑛 = 𝑁 → {𝑦 ∣ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))} = (𝑈↑↑𝑁))
7473eleq2d 2878 . . . . . . . . . . . . 13 (𝑛 = 𝑁 → (𝑦 ∈ {𝑦 ∣ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))} ↔ 𝑦 ∈ (𝑈↑↑𝑁)))
75 abid 2783 . . . . . . . . . . . . 13 (𝑦 ∈ {𝑦 ∣ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))} ↔ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)))
7674, 75bitr3di 289 . . . . . . . . . . . 12 (𝑛 = 𝑁 → (𝑦 ∈ (𝑈↑↑𝑁) ↔ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))))
7776adantr 484 . . . . . . . . . . 11 ((𝑛 = 𝑁 ∧ (𝑛 ∈ ω ∧ (1o𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈)))) → (𝑦 ∈ (𝑈↑↑𝑁) ↔ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))))
7863, 77mtbird 328 . . . . . . . . . 10 ((𝑛 = 𝑁 ∧ (𝑛 ∈ ω ∧ (1o𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈)))) → ¬ 𝑦 ∈ (𝑈↑↑𝑁))
7978ex 416 . . . . . . . . 9 (𝑛 = 𝑁 → ((𝑛 ∈ ω ∧ (1o𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈))) → ¬ 𝑦 ∈ (𝑈↑↑𝑁)))
804, 79syl5bi 245 . . . . . . . 8 (𝑛 = 𝑁 → (((𝑛 ∈ ω ∧ 1o𝑛) ∧ ¬ 𝑦 ∈ (V × 𝑈)) → ¬ 𝑦 ∈ (𝑈↑↑𝑁)))
8180expdimp 456 . . . . . . 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 3924 . . . . 5 ((𝑛 = 𝑁 ∧ (𝑛 ∈ ω ∧ 1o𝑛)) → (𝑈↑↑𝑁) ⊆ (V × 𝑈))
8483ex 416 . . . 4 (𝑛 = 𝑁 → ((𝑛 ∈ ω ∧ 1o𝑛) → (𝑈↑↑𝑁) ⊆ (V × 𝑈)))
853, 84sylbird 263 . . 3 (𝑛 = 𝑁 → ((𝑁 ∈ ω ∧ 1o𝑁) → (𝑈↑↑𝑁) ⊆ (V × 𝑈)))
8685vtocleg 3532 . 2 (𝑁 ∈ ω → ((𝑁 ∈ ω ∧ 1o𝑁) → (𝑈↑↑𝑁) ⊆ (V × 𝑈)))
8786anabsi5 668 1 ((𝑁 ∈ ω ∧ 1o𝑁) → (𝑈↑↑𝑁) ⊆ (V × 𝑈))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  {cab 2779  wne 2990  Vcvv 3444  wss 3884  c0 4246  ifcif 4428  cop 4534   cuni 4803   × cxp 5521  Oncon0 6163  suc csuc 6165  cfv 6328  cmpo 7141  ωcom 7564  1st c1st 7673  reccrdg 8032  1oc1o 8082  ↑↑cfinxp 34801
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-finxp 34802
This theorem is referenced by:  finxpsuclem  34815
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