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Theorem finxpreclem2 35792
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 1917 . . . . . 6 𝑥(𝑋 ∈ V ∧ ¬ 𝑋𝑈)
2 nfmpo2 7432 . . . . . . . 8 𝑥(𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
3 nfcv 2905 . . . . . . . 8 𝑥⟨1o, 𝑋
42, 3nffv 6849 . . . . . . 7 𝑥((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩)
5 nfcv 2905 . . . . . . 7 𝑥
64, 5nfne 3043 . . . . . 6 𝑥((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩) ≠ ∅
71, 6nfim 1899 . . . . 5 𝑥((𝑋 ∈ V ∧ ¬ 𝑋𝑈) → ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩) ≠ ∅)
8 nfv 1917 . . . . . . 7 𝑛 𝑥 = 𝑋
9 nfv 1917 . . . . . . . 8 𝑛(𝑋 ∈ V ∧ ¬ 𝑋𝑈)
10 nfmpo1 7431 . . . . . . . . . 10 𝑛(𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
11 nfcv 2905 . . . . . . . . . 10 𝑛⟨1o, 𝑋
1210, 11nffv 6849 . . . . . . . . 9 𝑛((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩)
13 nfcv 2905 . . . . . . . . 9 𝑛
1412, 13nfne 3043 . . . . . . . 8 𝑛((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩) ≠ ∅
159, 14nfim 1899 . . . . . . 7 𝑛((𝑋 ∈ V ∧ ¬ 𝑋𝑈) → ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩) ≠ ∅)
168, 15nfim 1899 . . . . . 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 8578 . . . . . . 7 1o ∈ ω
1817elexi 3462 . . . . . 6 1o ∈ V
19 df-ov 7354 . . . . . . . . . 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 5262 . . . . . . . . . . . . . . . 16 ∅ ∈ V
21 opex 5419 . . . . . . . . . . . . . . . . 17 𝑛, (1st𝑥)⟩ ∈ V
22 opex 5419 . . . . . . . . . . . . . . . . 17 𝑛, 𝑥⟩ ∈ V
2321, 22ifex 4534 . . . . . . . . . . . . . . . 16 if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) ∈ V
2420, 23ifex 4534 . . . . . . . . . . . . . . 15 if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) ∈ V
2524csbex 5266 . . . . . . . . . . . . . 14 𝑋 / 𝑥if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) ∈ V
2625csbex 5266 . . . . . . . . . . . . 13 1o / 𝑛𝑋 / 𝑥if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) ∈ V
27 eqid 2737 . . . . . . . . . . . . . 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 7497 . . . . . . . . . . . . 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 1452 . . . . . . . . . . . 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 481 . . . . . . . . . . 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 3867 . . . . . . . . . . . . . . 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 3867 . . . . . . . . . . . . . . 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 2799 . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . 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 2825 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑋 → (𝑥𝑈𝑋𝑈))
3635notbid 317 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑋 → (¬ 𝑥𝑈 ↔ ¬ 𝑋𝑈))
3736biimprcd 249 . . . . . . . . . . . . . . . . . 18 𝑋𝑈 → (𝑥 = 𝑋 → ¬ 𝑥𝑈))
38 pm3.14 994 . . . . . . . . . . . . . . . . . . 19 ((¬ 𝑛 = 1o ∨ ¬ 𝑥𝑈) → ¬ (𝑛 = 1o𝑥𝑈))
3938olcs 874 . . . . . . . . . . . . . . . . . 18 𝑥𝑈 → ¬ (𝑛 = 1o𝑥𝑈))
4037, 39syl6 35 . . . . . . . . . . . . . . . . 17 𝑋𝑈 → (𝑥 = 𝑋 → ¬ (𝑛 = 1o𝑥𝑈)))
41 iffalse 4493 . . . . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . . . 15 ((¬ 𝑋𝑈𝑥 = 𝑋) → if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) = if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))
44 ifeqor 4535 . . . . . . . . . . . . . . . . 17 (if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) = ⟨ 𝑛, (1st𝑥)⟩ ∨ if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩)
45 vuniex 7668 . . . . . . . . . . . . . . . . . . . . 21 𝑛 ∈ V
46 fvex 6852 . . . . . . . . . . . . . . . . . . . . 21 (1st𝑥) ∈ V
4745, 46opnzi 5429 . . . . . . . . . . . . . . . . . . . 20 𝑛, (1st𝑥)⟩ ≠ ∅
4847neii 2943 . . . . . . . . . . . . . . . . . . 19 ¬ ⟨ 𝑛, (1st𝑥)⟩ = ∅
49 eqeq1 2741 . . . . . . . . . . . . . . . . . . 19 (if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) = ⟨ 𝑛, (1st𝑥)⟩ → (if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) = ∅ ↔ ⟨ 𝑛, (1st𝑥)⟩ = ∅))
5048, 49mtbiri 326 . . . . . . . . . . . . . . . . . 18 (if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) = ⟨ 𝑛, (1st𝑥)⟩ → ¬ if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) = ∅)
51 vex 3447 . . . . . . . . . . . . . . . . . . . . 21 𝑛 ∈ V
52 vex 3447 . . . . . . . . . . . . . . . . . . . . 21 𝑥 ∈ V
5351, 52opnzi 5429 . . . . . . . . . . . . . . . . . . . 20 𝑛, 𝑥⟩ ≠ ∅
5453neii 2943 . . . . . . . . . . . . . . . . . . 19 ¬ ⟨𝑛, 𝑥⟩ = ∅
55 eqeq1 2741 . . . . . . . . . . . . . . . . . . 19 (if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩ → (if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) = ∅ ↔ ⟨𝑛, 𝑥⟩ = ∅))
5654, 55mtbiri 326 . . . . . . . . . . . . . . . . . 18 (if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩ → ¬ if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) = ∅)
5750, 56jaoi 855 . . . . . . . . . . . . . . . . 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 714 . . . . . . . . . . . . 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 482 . . . . . . . . . . 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 648 . . . . . . . 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 647 . . . . . . 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 420 . . . . . 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 3513 . . . . 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 35736 . . . 4 (𝑋 ∈ V → ((𝑋 ∈ V ∧ ¬ 𝑋𝑈) → ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩) ≠ ∅))
7170anabsi5 667 . . 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 396  wo 845   = wceq 1541  wcel 2106  wne 2941  Vcvv 3443  csb 3853  c0 4280  ifcif 4484  cop 4590   cuni 4863   × cxp 5629  cfv 6493  (class class class)co 7351  cmpo 7353  ωcom 7794  1st c1st 7911  1oc1o 8397
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-tr 5221  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6445  df-fun 6495  df-fv 6501  df-ov 7354  df-oprab 7355  df-mpo 7356  df-om 7795  df-1o 8404
This theorem is referenced by:  finxp1o  35794
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