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Theorem finxpreclem2 33538
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 2005 . . . . . 6 𝑥(𝑋 ∈ V ∧ ¬ 𝑋𝑈)
2 nfmpt22 6949 . . . . . . . 8 𝑥(𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
3 nfcv 2948 . . . . . . . 8 𝑥⟨1𝑜, 𝑋
42, 3nffv 6414 . . . . . . 7 𝑥((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1𝑜, 𝑋⟩)
5 nfcv 2948 . . . . . . 7 𝑥
64, 5nfne 3078 . . . . . 6 𝑥((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1𝑜, 𝑋⟩) ≠ ∅
71, 6nfim 1987 . . . . 5 𝑥((𝑋 ∈ V ∧ ¬ 𝑋𝑈) → ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1𝑜, 𝑋⟩) ≠ ∅)
8 nfv 2005 . . . . . . 7 𝑛 𝑥 = 𝑋
9 nfv 2005 . . . . . . . 8 𝑛(𝑋 ∈ V ∧ ¬ 𝑋𝑈)
10 nfmpt21 6948 . . . . . . . . . 10 𝑛(𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
11 nfcv 2948 . . . . . . . . . 10 𝑛⟨1𝑜, 𝑋
1210, 11nffv 6414 . . . . . . . . 9 𝑛((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1𝑜, 𝑋⟩)
13 nfcv 2948 . . . . . . . . 9 𝑛
1412, 13nfne 3078 . . . . . . . 8 𝑛((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1𝑜, 𝑋⟩) ≠ ∅
159, 14nfim 1987 . . . . . . 7 𝑛((𝑋 ∈ V ∧ ¬ 𝑋𝑈) → ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1𝑜, 𝑋⟩) ≠ ∅)
168, 15nfim 1987 . . . . . 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 7952 . . . . . . 7 1𝑜 ∈ ω
1817elexi 3407 . . . . . 6 1𝑜 ∈ V
19 df-ov 6873 . . . . . . . . . 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 4984 . . . . . . . . . . . . . . . 16 ∅ ∈ V
21 opex 5122 . . . . . . . . . . . . . . . . 17 𝑛, (1st𝑥)⟩ ∈ V
22 opex 5122 . . . . . . . . . . . . . . . . 17 𝑛, 𝑥⟩ ∈ V
2321, 22ifex 4327 . . . . . . . . . . . . . . . 16 if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) ∈ V
2420, 23ifex 4327 . . . . . . . . . . . . . . 15 if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) ∈ V
2524csbex 4988 . . . . . . . . . . . . . 14 𝑋 / 𝑥if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) ∈ V
2625csbex 4988 . . . . . . . . . . . . 13 1𝑜 / 𝑛𝑋 / 𝑥if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) ∈ V
27 eqid 2806 . . . . . . . . . . . . . 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⟩)))
2827ovmpt2s 7010 . . . . . . . . . . . . 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 1569 . . . . . . . . . . . 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 468 . . . . . . . . . . 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 3737 . . . . . . . . . . . . . . 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 3737 . . . . . . . . . . . . . . 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 2862 . . . . . . . . . . . . . 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 469 . . . . . . . . . . . . 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 2873 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑋 → (𝑥𝑈𝑋𝑈))
3635notbid 309 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑋 → (¬ 𝑥𝑈 ↔ ¬ 𝑋𝑈))
3736biimprcd 241 . . . . . . . . . . . . . . . . . 18 𝑋𝑈 → (𝑥 = 𝑋 → ¬ 𝑥𝑈))
38 pm3.14 1009 . . . . . . . . . . . . . . . . . . 19 ((¬ 𝑛 = 1𝑜 ∨ ¬ 𝑥𝑈) → ¬ (𝑛 = 1𝑜𝑥𝑈))
3938olcs 894 . . . . . . . . . . . . . . . . . 18 𝑥𝑈 → ¬ (𝑛 = 1𝑜𝑥𝑈))
4037, 39syl6 35 . . . . . . . . . . . . . . . . 17 𝑋𝑈 → (𝑥 = 𝑋 → ¬ (𝑛 = 1𝑜𝑥𝑈)))
41 iffalse 4288 . . . . . . . . . . . . . . . . 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 395 . . . . . . . . . . . . . . 15 ((¬ 𝑋𝑈𝑥 = 𝑋) → if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) = if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))
44 ifeqor 4328 . . . . . . . . . . . . . . . . 17 (if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) = ⟨ 𝑛, (1st𝑥)⟩ ∨ if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩)
45 vuniex 7180 . . . . . . . . . . . . . . . . . . . . 21 𝑛 ∈ V
46 fvex 6417 . . . . . . . . . . . . . . . . . . . . 21 (1st𝑥) ∈ V
4745, 46opnzi 5132 . . . . . . . . . . . . . . . . . . . 20 𝑛, (1st𝑥)⟩ ≠ ∅
4847neii 2980 . . . . . . . . . . . . . . . . . . 19 ¬ ⟨ 𝑛, (1st𝑥)⟩ = ∅
49 eqeq1 2810 . . . . . . . . . . . . . . . . . . 19 (if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) = ⟨ 𝑛, (1st𝑥)⟩ → (if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) = ∅ ↔ ⟨ 𝑛, (1st𝑥)⟩ = ∅))
5048, 49mtbiri 318 . . . . . . . . . . . . . . . . . 18 (if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) = ⟨ 𝑛, (1st𝑥)⟩ → ¬ if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) = ∅)
51 vex 3394 . . . . . . . . . . . . . . . . . . . . 21 𝑛 ∈ V
52 vex 3394 . . . . . . . . . . . . . . . . . . . . 21 𝑥 ∈ V
5351, 52opnzi 5132 . . . . . . . . . . . . . . . . . . . 20 𝑛, 𝑥⟩ ≠ ∅
5453neii 2980 . . . . . . . . . . . . . . . . . . 19 ¬ ⟨𝑛, 𝑥⟩ = ∅
55 eqeq1 2810 . . . . . . . . . . . . . . . . . . 19 (if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩ → (if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) = ∅ ↔ ⟨𝑛, 𝑥⟩ = ∅))
5654, 55mtbiri 318 . . . . . . . . . . . . . . . . . 18 (if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩ → ¬ if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) = ∅)
5750, 56jaoi 875 . . . . . . . . . . . . . . . . 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 2985 . . . . . . . . . . . . . . 15 ((¬ 𝑋𝑈𝑥 = 𝑋) → if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) ≠ ∅)
6043, 59eqnetrd 3045 . . . . . . . . . . . . . 14 ((¬ 𝑋𝑈𝑥 = 𝑋) → if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) ≠ ∅)
6160adantrl 698 . . . . . . . . . . . . 13 ((¬ 𝑋𝑈 ∧ (𝑛 = 1𝑜𝑥 = 𝑋)) → if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) ≠ ∅)
6234, 61eqnetrrd 3046 . . . . . . . . . . . 12 ((¬ 𝑋𝑈 ∧ (𝑛 = 1𝑜𝑥 = 𝑋)) → 1𝑜 / 𝑛𝑋 / 𝑥if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) ≠ ∅)
6362adantl 469 . . . . . . . . . . 11 ((𝑋 ∈ V ∧ (¬ 𝑋𝑈 ∧ (𝑛 = 1𝑜𝑥 = 𝑋))) → 1𝑜 / 𝑛𝑋 / 𝑥if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) ≠ ∅)
6430, 63eqnetrd 3045 . . . . . . . . . 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, 64syl5eqner 3053 . . . . . . . . 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 632 . . . . . . . 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 631 . . . . . . 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 408 . . . . . 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 3474 . . . . 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 33493 . . . 4 (𝑋 ∈ V → ((𝑋 ∈ V ∧ ¬ 𝑋𝑈) → ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1𝑜, 𝑋⟩) ≠ ∅))
7170anabsi5 651 . . 3 ((𝑋 ∈ V ∧ ¬ 𝑋𝑈) → ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1𝑜, 𝑋⟩) ≠ ∅)
7271necomd 3033 . 2 ((𝑋 ∈ V ∧ ¬ 𝑋𝑈) → ∅ ≠ ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1𝑜, 𝑋⟩))
7372neneqd 2983 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 384  wo 865   = wceq 1637  wcel 2156  wne 2978  Vcvv 3391  csb 3728  c0 4116  ifcif 4279  cop 4376   cuni 4630   × cxp 5309  cfv 6097  (class class class)co 6870  cmpt2 6872  ωcom 7291  1st c1st 7392  1𝑜c1o 7785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pr 5096  ax-un 7175
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6060  df-fun 6099  df-fv 6105  df-ov 6873  df-oprab 6874  df-mpt2 6875  df-om 7292  df-1o 7792
This theorem is referenced by:  finxp1o  33540
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