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Theorem finxpreclem2 37373
Description: Lemma for ↑↑ recursion theorems. (Contributed by ML, 17-Oct-2020.)
Assertion
Ref Expression
finxpreclem2 ((𝑋 ∈ V ∧ ¬ 𝑋𝑈) → ¬ ∅ = ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩))
Distinct variable groups:   𝑈,𝑛,𝑥   𝑛,𝑋,𝑥

Proof of Theorem finxpreclem2
StepHypRef Expression
1 nfv 1912 . . . . . 6 𝑥(𝑋 ∈ V ∧ ¬ 𝑋𝑈)
2 nfmpo2 7514 . . . . . . . 8 𝑥(𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
3 nfcv 2903 . . . . . . . 8 𝑥⟨1o, 𝑋
42, 3nffv 6917 . . . . . . 7 𝑥((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩)
5 nfcv 2903 . . . . . . 7 𝑥
64, 5nfne 3041 . . . . . 6 𝑥((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩) ≠ ∅
71, 6nfim 1894 . . . . 5 𝑥((𝑋 ∈ V ∧ ¬ 𝑋𝑈) → ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩) ≠ ∅)
8 nfv 1912 . . . . . . 7 𝑛 𝑥 = 𝑋
9 nfv 1912 . . . . . . . 8 𝑛(𝑋 ∈ V ∧ ¬ 𝑋𝑈)
10 nfmpo1 7513 . . . . . . . . . 10 𝑛(𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
11 nfcv 2903 . . . . . . . . . 10 𝑛⟨1o, 𝑋
1210, 11nffv 6917 . . . . . . . . 9 𝑛((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩)
13 nfcv 2903 . . . . . . . . 9 𝑛
1412, 13nfne 3041 . . . . . . . 8 𝑛((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩) ≠ ∅
159, 14nfim 1894 . . . . . . 7 𝑛((𝑋 ∈ V ∧ ¬ 𝑋𝑈) → ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩) ≠ ∅)
168, 15nfim 1894 . . . . . 6 𝑛(𝑥 = 𝑋 → ((𝑋 ∈ V ∧ ¬ 𝑋𝑈) → ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩) ≠ ∅))
17 1onn 8677 . . . . . . 7 1o ∈ ω
1817elexi 3501 . . . . . 6 1o ∈ V
19 df-ov 7434 . . . . . . . . . 10 (1o(𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))𝑋) = ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩)
20 0ex 5313 . . . . . . . . . . . . . . . 16 ∅ ∈ V
21 opex 5475 . . . . . . . . . . . . . . . . 17 𝑛, (1st𝑥)⟩ ∈ V
22 opex 5475 . . . . . . . . . . . . . . . . 17 𝑛, 𝑥⟩ ∈ V
2321, 22ifex 4581 . . . . . . . . . . . . . . . 16 if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) ∈ V
2420, 23ifex 4581 . . . . . . . . . . . . . . 15 if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) ∈ V
2524csbex 5317 . . . . . . . . . . . . . 14 𝑋 / 𝑥if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) ∈ V
2625csbex 5317 . . . . . . . . . . . . 13 1o / 𝑛𝑋 / 𝑥if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) ∈ V
27 eqid 2735 . . . . . . . . . . . . . 14 (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))) = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
2827ovmpos 7581 . . . . . . . . . . . . 13 ((1o ∈ ω ∧ 𝑋 ∈ V ∧ 1o / 𝑛𝑋 / 𝑥if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) ∈ V) → (1o(𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))𝑋) = 1o / 𝑛𝑋 / 𝑥if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
2917, 26, 28mp3an13 1451 . . . . . . . . . . . 12 (𝑋 ∈ V → (1o(𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))𝑋) = 1o / 𝑛𝑋 / 𝑥if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
3029adantr 480 . . . . . . . . . . 11 ((𝑋 ∈ V ∧ (¬ 𝑋𝑈 ∧ (𝑛 = 1o𝑥 = 𝑋))) → (1o(𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))𝑋) = 1o / 𝑛𝑋 / 𝑥if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
31 csbeq1a 3922 . . . . . . . . . . . . . . 15 (𝑥 = 𝑋 → if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) = 𝑋 / 𝑥if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
32 csbeq1a 3922 . . . . . . . . . . . . . . 15 (𝑛 = 1o𝑋 / 𝑥if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) = 1o / 𝑛𝑋 / 𝑥if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
3331, 32sylan9eqr 2797 . . . . . . . . . . . . . 14 ((𝑛 = 1o𝑥 = 𝑋) → if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) = 1o / 𝑛𝑋 / 𝑥if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
3433adantl 481 . . . . . . . . . . . . 13 ((¬ 𝑋𝑈 ∧ (𝑛 = 1o𝑥 = 𝑋)) → if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) = 1o / 𝑛𝑋 / 𝑥if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
35 eleq1 2827 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑋 → (𝑥𝑈𝑋𝑈))
3635notbid 318 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑋 → (¬ 𝑥𝑈 ↔ ¬ 𝑋𝑈))
3736biimprcd 250 . . . . . . . . . . . . . . . . . 18 𝑋𝑈 → (𝑥 = 𝑋 → ¬ 𝑥𝑈))
38 pm3.14 997 . . . . . . . . . . . . . . . . . . 19 ((¬ 𝑛 = 1o ∨ ¬ 𝑥𝑈) → ¬ (𝑛 = 1o𝑥𝑈))
3938olcs 876 . . . . . . . . . . . . . . . . . 18 𝑥𝑈 → ¬ (𝑛 = 1o𝑥𝑈))
4037, 39syl6 35 . . . . . . . . . . . . . . . . 17 𝑋𝑈 → (𝑥 = 𝑋 → ¬ (𝑛 = 1o𝑥𝑈)))
41 iffalse 4540 . . . . . . . . . . . . . . . . 17 (¬ (𝑛 = 1o𝑥𝑈) → if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) = if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))
4240, 41syl6 35 . . . . . . . . . . . . . . . 16 𝑋𝑈 → (𝑥 = 𝑋 → if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) = if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
4342imp 406 . . . . . . . . . . . . . . 15 ((¬ 𝑋𝑈𝑥 = 𝑋) → if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) = if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))
44 ifeqor 4582 . . . . . . . . . . . . . . . . 17 (if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) = ⟨ 𝑛, (1st𝑥)⟩ ∨ if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩)
45 vuniex 7758 . . . . . . . . . . . . . . . . . . . . 21 𝑛 ∈ V
46 fvex 6920 . . . . . . . . . . . . . . . . . . . . 21 (1st𝑥) ∈ V
4745, 46opnzi 5485 . . . . . . . . . . . . . . . . . . . 20 𝑛, (1st𝑥)⟩ ≠ ∅
4847neii 2940 . . . . . . . . . . . . . . . . . . 19 ¬ ⟨ 𝑛, (1st𝑥)⟩ = ∅
49 eqeq1 2739 . . . . . . . . . . . . . . . . . . 19 (if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) = ⟨ 𝑛, (1st𝑥)⟩ → (if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) = ∅ ↔ ⟨ 𝑛, (1st𝑥)⟩ = ∅))
5048, 49mtbiri 327 . . . . . . . . . . . . . . . . . 18 (if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) = ⟨ 𝑛, (1st𝑥)⟩ → ¬ if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) = ∅)
51 vex 3482 . . . . . . . . . . . . . . . . . . . . 21 𝑛 ∈ V
52 vex 3482 . . . . . . . . . . . . . . . . . . . . 21 𝑥 ∈ V
5351, 52opnzi 5485 . . . . . . . . . . . . . . . . . . . 20 𝑛, 𝑥⟩ ≠ ∅
5453neii 2940 . . . . . . . . . . . . . . . . . . 19 ¬ ⟨𝑛, 𝑥⟩ = ∅
55 eqeq1 2739 . . . . . . . . . . . . . . . . . . 19 (if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩ → (if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) = ∅ ↔ ⟨𝑛, 𝑥⟩ = ∅))
5654, 55mtbiri 327 . . . . . . . . . . . . . . . . . 18 (if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩ → ¬ if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) = ∅)
5750, 56jaoi 857 . . . . . . . . . . . . . . . . 17 ((if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) = ⟨ 𝑛, (1st𝑥)⟩ ∨ if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩) → ¬ if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) = ∅)
5844, 57mp1i 13 . . . . . . . . . . . . . . . 16 ((¬ 𝑋𝑈𝑥 = 𝑋) → ¬ if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) = ∅)
5958neqned 2945 . . . . . . . . . . . . . . 15 ((¬ 𝑋𝑈𝑥 = 𝑋) → if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) ≠ ∅)
6043, 59eqnetrd 3006 . . . . . . . . . . . . . 14 ((¬ 𝑋𝑈𝑥 = 𝑋) → if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) ≠ ∅)
6160adantrl 716 . . . . . . . . . . . . 13 ((¬ 𝑋𝑈 ∧ (𝑛 = 1o𝑥 = 𝑋)) → if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) ≠ ∅)
6234, 61eqnetrrd 3007 . . . . . . . . . . . 12 ((¬ 𝑋𝑈 ∧ (𝑛 = 1o𝑥 = 𝑋)) → 1o / 𝑛𝑋 / 𝑥if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) ≠ ∅)
6362adantl 481 . . . . . . . . . . 11 ((𝑋 ∈ V ∧ (¬ 𝑋𝑈 ∧ (𝑛 = 1o𝑥 = 𝑋))) → 1o / 𝑛𝑋 / 𝑥if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) ≠ ∅)
6430, 63eqnetrd 3006 . . . . . . . . . 10 ((𝑋 ∈ V ∧ (¬ 𝑋𝑈 ∧ (𝑛 = 1o𝑥 = 𝑋))) → (1o(𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))𝑋) ≠ ∅)
6519, 64eqnetrrid 3014 . . . . . . . . 9 ((𝑋 ∈ V ∧ (¬ 𝑋𝑈 ∧ (𝑛 = 1o𝑥 = 𝑋))) → ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩) ≠ ∅)
6665ancom2s 650 . . . . . . . 8 ((𝑋 ∈ V ∧ ((𝑛 = 1o𝑥 = 𝑋) ∧ ¬ 𝑋𝑈)) → ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩) ≠ ∅)
6766an12s 649 . . . . . . 7 (((𝑛 = 1o𝑥 = 𝑋) ∧ (𝑋 ∈ V ∧ ¬ 𝑋𝑈)) → ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩) ≠ ∅)
6867exp31 419 . . . . . 6 (𝑛 = 1o → (𝑥 = 𝑋 → ((𝑋 ∈ V ∧ ¬ 𝑋𝑈) → ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩) ≠ ∅)))
6916, 18, 68vtoclef 3563 . . . . 5 (𝑥 = 𝑋 → ((𝑋 ∈ V ∧ ¬ 𝑋𝑈) → ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩) ≠ ∅))
707, 69vtoclefex 37317 . . . 4 (𝑋 ∈ V → ((𝑋 ∈ V ∧ ¬ 𝑋𝑈) → ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩) ≠ ∅))
7170anabsi5 669 . . 3 ((𝑋 ∈ V ∧ ¬ 𝑋𝑈) → ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩) ≠ ∅)
7271necomd 2994 . 2 ((𝑋 ∈ V ∧ ¬ 𝑋𝑈) → ∅ ≠ ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩))
7372neneqd 2943 1 ((𝑋 ∈ V ∧ ¬ 𝑋𝑈) → ¬ ∅ = ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 847   = wceq 1537  wcel 2106  wne 2938  Vcvv 3478  csb 3908  c0 4339  ifcif 4531  cop 4637   cuni 4912   × cxp 5687  cfv 6563  (class class class)co 7431  cmpo 7433  ωcom 7887  1st c1st 8011  1oc1o 8498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1o 8505
This theorem is referenced by:  finxp1o  37375
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