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Theorem finxpreclem2 36271
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 1918 . . . . . 6 𝑥(𝑋 ∈ V ∧ ¬ 𝑋𝑈)
2 nfmpo2 7490 . . . . . . . 8 𝑥(𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
3 nfcv 2904 . . . . . . . 8 𝑥⟨1o, 𝑋
42, 3nffv 6902 . . . . . . 7 𝑥((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩)
5 nfcv 2904 . . . . . . 7 𝑥
64, 5nfne 3044 . . . . . 6 𝑥((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩) ≠ ∅
71, 6nfim 1900 . . . . 5 𝑥((𝑋 ∈ V ∧ ¬ 𝑋𝑈) → ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩) ≠ ∅)
8 nfv 1918 . . . . . . 7 𝑛 𝑥 = 𝑋
9 nfv 1918 . . . . . . . 8 𝑛(𝑋 ∈ V ∧ ¬ 𝑋𝑈)
10 nfmpo1 7489 . . . . . . . . . 10 𝑛(𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
11 nfcv 2904 . . . . . . . . . 10 𝑛⟨1o, 𝑋
1210, 11nffv 6902 . . . . . . . . 9 𝑛((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩)
13 nfcv 2904 . . . . . . . . 9 𝑛
1412, 13nfne 3044 . . . . . . . 8 𝑛((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩) ≠ ∅
159, 14nfim 1900 . . . . . . 7 𝑛((𝑋 ∈ V ∧ ¬ 𝑋𝑈) → ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩) ≠ ∅)
168, 15nfim 1900 . . . . . 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 8639 . . . . . . 7 1o ∈ ω
1817elexi 3494 . . . . . 6 1o ∈ V
19 df-ov 7412 . . . . . . . . . 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 5308 . . . . . . . . . . . . . . . 16 ∅ ∈ V
21 opex 5465 . . . . . . . . . . . . . . . . 17 𝑛, (1st𝑥)⟩ ∈ V
22 opex 5465 . . . . . . . . . . . . . . . . 17 𝑛, 𝑥⟩ ∈ V
2321, 22ifex 4579 . . . . . . . . . . . . . . . 16 if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) ∈ V
2420, 23ifex 4579 . . . . . . . . . . . . . . 15 if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) ∈ V
2524csbex 5312 . . . . . . . . . . . . . 14 𝑋 / 𝑥if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) ∈ V
2625csbex 5312 . . . . . . . . . . . . 13 1o / 𝑛𝑋 / 𝑥if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) ∈ V
27 eqid 2733 . . . . . . . . . . . . . 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 7556 . . . . . . . . . . . . 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 1453 . . . . . . . . . . . 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 482 . . . . . . . . . . 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 3908 . . . . . . . . . . . . . . 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 3908 . . . . . . . . . . . . . . 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 2795 . . . . . . . . . . . . . 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 483 . . . . . . . . . . . . 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 2822 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑋 → (𝑥𝑈𝑋𝑈))
3635notbid 318 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑋 → (¬ 𝑥𝑈 ↔ ¬ 𝑋𝑈))
3736biimprcd 249 . . . . . . . . . . . . . . . . . 18 𝑋𝑈 → (𝑥 = 𝑋 → ¬ 𝑥𝑈))
38 pm3.14 995 . . . . . . . . . . . . . . . . . . 19 ((¬ 𝑛 = 1o ∨ ¬ 𝑥𝑈) → ¬ (𝑛 = 1o𝑥𝑈))
3938olcs 875 . . . . . . . . . . . . . . . . . 18 𝑥𝑈 → ¬ (𝑛 = 1o𝑥𝑈))
4037, 39syl6 35 . . . . . . . . . . . . . . . . 17 𝑋𝑈 → (𝑥 = 𝑋 → ¬ (𝑛 = 1o𝑥𝑈)))
41 iffalse 4538 . . . . . . . . . . . . . . . . 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 408 . . . . . . . . . . . . . . 15 ((¬ 𝑋𝑈𝑥 = 𝑋) → if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) = if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))
44 ifeqor 4580 . . . . . . . . . . . . . . . . 17 (if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) = ⟨ 𝑛, (1st𝑥)⟩ ∨ if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩)
45 vuniex 7729 . . . . . . . . . . . . . . . . . . . . 21 𝑛 ∈ V
46 fvex 6905 . . . . . . . . . . . . . . . . . . . . 21 (1st𝑥) ∈ V
4745, 46opnzi 5475 . . . . . . . . . . . . . . . . . . . 20 𝑛, (1st𝑥)⟩ ≠ ∅
4847neii 2943 . . . . . . . . . . . . . . . . . . 19 ¬ ⟨ 𝑛, (1st𝑥)⟩ = ∅
49 eqeq1 2737 . . . . . . . . . . . . . . . . . . 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 3479 . . . . . . . . . . . . . . . . . . . . 21 𝑛 ∈ V
52 vex 3479 . . . . . . . . . . . . . . . . . . . . 21 𝑥 ∈ V
5351, 52opnzi 5475 . . . . . . . . . . . . . . . . . . . 20 𝑛, 𝑥⟩ ≠ ∅
5453neii 2943 . . . . . . . . . . . . . . . . . . 19 ¬ ⟨𝑛, 𝑥⟩ = ∅
55 eqeq1 2737 . . . . . . . . . . . . . . . . . . 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 856 . . . . . . . . . . . . . . . . 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 2948 . . . . . . . . . . . . . . 15 ((¬ 𝑋𝑈𝑥 = 𝑋) → if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) ≠ ∅)
6043, 59eqnetrd 3009 . . . . . . . . . . . . . 14 ((¬ 𝑋𝑈𝑥 = 𝑋) → if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) ≠ ∅)
6160adantrl 715 . . . . . . . . . . . . 13 ((¬ 𝑋𝑈 ∧ (𝑛 = 1o𝑥 = 𝑋)) → if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) ≠ ∅)
6234, 61eqnetrrd 3010 . . . . . . . . . . . 12 ((¬ 𝑋𝑈 ∧ (𝑛 = 1o𝑥 = 𝑋)) → 1o / 𝑛𝑋 / 𝑥if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) ≠ ∅)
6362adantl 483 . . . . . . . . . . 11 ((𝑋 ∈ V ∧ (¬ 𝑋𝑈 ∧ (𝑛 = 1o𝑥 = 𝑋))) → 1o / 𝑛𝑋 / 𝑥if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) ≠ ∅)
6430, 63eqnetrd 3009 . . . . . . . . . 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 3017 . . . . . . . . 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 649 . . . . . . . 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 648 . . . . . . 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 421 . . . . . 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 3547 . . . . 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 36215 . . . 4 (𝑋 ∈ V → ((𝑋 ∈ V ∧ ¬ 𝑋𝑈) → ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩) ≠ ∅))
7170anabsi5 668 . . 3 ((𝑋 ∈ V ∧ ¬ 𝑋𝑈) → ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩) ≠ ∅)
7271necomd 2997 . 2 ((𝑋 ∈ V ∧ ¬ 𝑋𝑈) → ∅ ≠ ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩))
7372neneqd 2946 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 397  wo 846   = wceq 1542  wcel 2107  wne 2941  Vcvv 3475  csb 3894  c0 4323  ifcif 4529  cop 4635   cuni 4909   × cxp 5675  cfv 6544  (class class class)co 7409  cmpo 7411  ωcom 7855  1st c1st 7973  1oc1o 8459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1o 8466
This theorem is referenced by:  finxp1o  36273
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