Theorem finxpreclem2 37595

Description: Lemma for ↑↑ recursion theorems. (Contributed by ML, 17-Oct-2020.)

Assertion

Ref	Expression
finxpreclem2	⊢ ((𝑋 ∈ V ∧ ¬ 𝑋 ∈ 𝑈) → ¬ ∅ = ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩))

Distinct variable groups:   𝑈,𝑛,𝑥   𝑛,𝑋,𝑥

Proof of Theorem finxpreclem2

Step	Hyp	Ref	Expression
1		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑥(𝑋 ∈ V ∧ ¬ 𝑋 ∈ 𝑈)
2		nfmpo2 7439	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))
3		nfcv 2898	. . . . . . . 8 ⊢ Ⅎ𝑥⟨1o, 𝑋⟩
4	2, 3	nffv 6844	. . . . . . 7 ⊢ Ⅎ𝑥((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩)
5		nfcv 2898	. . . . . . 7 ⊢ Ⅎ𝑥∅
6	4, 5	nfne 3033	. . . . . 6 ⊢ Ⅎ𝑥((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩) ≠ ∅
7	1, 6	nfim 1897	. . . . 5 ⊢ Ⅎ𝑥((𝑋 ∈ V ∧ ¬ 𝑋 ∈ 𝑈) → ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩) ≠ ∅)
8		nfv 1915	. . . . . . 7 ⊢ Ⅎ𝑛 𝑥 = 𝑋
9		nfv 1915	. . . . . . . 8 ⊢ Ⅎ𝑛(𝑋 ∈ V ∧ ¬ 𝑋 ∈ 𝑈)
10		nfmpo1 7438	. . . . . . . . . 10 ⊢ Ⅎ𝑛(𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))
11		nfcv 2898	. . . . . . . . . 10 ⊢ Ⅎ𝑛⟨1o, 𝑋⟩
12	10, 11	nffv 6844	. . . . . . . . 9 ⊢ Ⅎ𝑛((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩)
13		nfcv 2898	. . . . . . . . 9 ⊢ Ⅎ𝑛∅
14	12, 13	nfne 3033	. . . . . . . 8 ⊢ Ⅎ𝑛((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩) ≠ ∅
15	9, 14	nfim 1897	. . . . . . 7 ⊢ Ⅎ𝑛((𝑋 ∈ V ∧ ¬ 𝑋 ∈ 𝑈) → ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩) ≠ ∅)
16	8, 15	nfim 1897	. . . . . 6 ⊢ Ⅎ𝑛(𝑥 = 𝑋 → ((𝑋 ∈ V ∧ ¬ 𝑋 ∈ 𝑈) → ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩) ≠ ∅))
17		1onn 8568	. . . . . . 7 ⊢ 1o ∈ ω
18	17	elexi 3463	. . . . . 6 ⊢ 1o ∈ V
19		df-ov 7361	. . . . . . . . . 10 ⊢ (1o(𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))𝑋) = ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩)
20		0ex 5252	. . . . . . . . . . . . . . . 16 ⊢ ∅ ∈ V
21		opex 5412	. . . . . . . . . . . . . . . . 17 ⊢ ⟨∪ 𝑛, (1st ‘𝑥)⟩ ∈ V
22		opex 5412	. . . . . . . . . . . . . . . . 17 ⊢ ⟨𝑛, 𝑥⟩ ∈ V
23	21, 22	ifex 4530	. . . . . . . . . . . . . . . 16 ⊢ if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩) ∈ V
24	20, 23	ifex 4530	. . . . . . . . . . . . . . 15 ⊢ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)) ∈ V
25	24	csbex 5256	. . . . . . . . . . . . . 14 ⊢ ⦋𝑋 / 𝑥⦌if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)) ∈ V
26	25	csbex 5256	. . . . . . . . . . . . 13 ⊢ ⦋1o / 𝑛⦌⦋𝑋 / 𝑥⦌if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)) ∈ V
27		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))) = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))
28	27	ovmpos 7506	. . . . . . . . . . . . 13 ⊢ ((1o ∈ ω ∧ 𝑋 ∈ V ∧ ⦋1o / 𝑛⦌⦋𝑋 / 𝑥⦌if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)) ∈ V) → (1o(𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))𝑋) = ⦋1o / 𝑛⦌⦋𝑋 / 𝑥⦌if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))
29	17, 26, 28	mp3an13 1454	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ V → (1o(𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))𝑋) = ⦋1o / 𝑛⦌⦋𝑋 / 𝑥⦌if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))
30	29	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ V ∧ (¬ 𝑋 ∈ 𝑈 ∧ (𝑛 = 1o ∧ 𝑥 = 𝑋))) → (1o(𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))𝑋) = ⦋1o / 𝑛⦌⦋𝑋 / 𝑥⦌if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))
31		csbeq1a 3863	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑋 → if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)) = ⦋𝑋 / 𝑥⦌if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))
32		csbeq1a 3863	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 1o → ⦋𝑋 / 𝑥⦌if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)) = ⦋1o / 𝑛⦌⦋𝑋 / 𝑥⦌if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))
33	31, 32	sylan9eqr 2793	. . . . . . . . . . . . . 14 ⊢ ((𝑛 = 1o ∧ 𝑥 = 𝑋) → if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)) = ⦋1o / 𝑛⦌⦋𝑋 / 𝑥⦌if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))
34	33	adantl 481	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑋 ∈ 𝑈 ∧ (𝑛 = 1o ∧ 𝑥 = 𝑋)) → if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)) = ⦋1o / 𝑛⦌⦋𝑋 / 𝑥⦌if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))
35		eleq1 2824	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝑈 ↔ 𝑋 ∈ 𝑈))
36	35	notbid 318	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑋 → (¬ 𝑥 ∈ 𝑈 ↔ ¬ 𝑋 ∈ 𝑈))
37	36	biimprcd 250	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ 𝑋 ∈ 𝑈 → (𝑥 = 𝑋 → ¬ 𝑥 ∈ 𝑈))
38		pm3.14 997	. . . . . . . . . . . . . . . . . . 19 ⊢ ((¬ 𝑛 = 1o ∨ ¬ 𝑥 ∈ 𝑈) → ¬ (𝑛 = 1o ∧ 𝑥 ∈ 𝑈))
39	38	olcs 876	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ 𝑥 ∈ 𝑈 → ¬ (𝑛 = 1o ∧ 𝑥 ∈ 𝑈))
40	37, 39	syl6 35	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑋 ∈ 𝑈 → (𝑥 = 𝑋 → ¬ (𝑛 = 1o ∧ 𝑥 ∈ 𝑈)))
41		iffalse 4488	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (𝑛 = 1o ∧ 𝑥 ∈ 𝑈) → if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)) = if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))
42	40, 41	syl6 35	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑋 ∈ 𝑈 → (𝑥 = 𝑋 → if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)) = if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))
43	42	imp 406	. . . . . . . . . . . . . . 15 ⊢ ((¬ 𝑋 ∈ 𝑈 ∧ 𝑥 = 𝑋) → if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)) = if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))
44		ifeqor 4531	. . . . . . . . . . . . . . . . 17 ⊢ (if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩) = ⟨∪ 𝑛, (1st ‘𝑥)⟩ ∨ if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩)
45		vuniex 7684	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ∪ 𝑛 ∈ V
46		fvex 6847	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1st ‘𝑥) ∈ V
47	45, 46	opnzi 5422	. . . . . . . . . . . . . . . . . . . 20 ⊢ ⟨∪ 𝑛, (1st ‘𝑥)⟩ ≠ ∅
48	47	neii 2934	. . . . . . . . . . . . . . . . . . 19 ⊢ ¬ ⟨∪ 𝑛, (1st ‘𝑥)⟩ = ∅
49		eqeq1 2740	. . . . . . . . . . . . . . . . . . 19 ⊢ (if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩) = ⟨∪ 𝑛, (1st ‘𝑥)⟩ → (if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩) = ∅ ↔ ⟨∪ 𝑛, (1st ‘𝑥)⟩ = ∅))
50	48, 49	mtbiri 327	. . . . . . . . . . . . . . . . . 18 ⊢ (if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩) = ⟨∪ 𝑛, (1st ‘𝑥)⟩ → ¬ if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩) = ∅)
51		vex 3444	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑛 ∈ V
52		vex 3444	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑥 ∈ V
53	51, 52	opnzi 5422	. . . . . . . . . . . . . . . . . . . 20 ⊢ ⟨𝑛, 𝑥⟩ ≠ ∅
54	53	neii 2934	. . . . . . . . . . . . . . . . . . 19 ⊢ ¬ ⟨𝑛, 𝑥⟩ = ∅
55		eqeq1 2740	. . . . . . . . . . . . . . . . . . 19 ⊢ (if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩ → (if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩) = ∅ ↔ ⟨𝑛, 𝑥⟩ = ∅))
56	54, 55	mtbiri 327	. . . . . . . . . . . . . . . . . 18 ⊢ (if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩ → ¬ if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩) = ∅)
57	50, 56	jaoi 857	. . . . . . . . . . . . . . . . 17 ⊢ ((if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩) = ⟨∪ 𝑛, (1st ‘𝑥)⟩ ∨ if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩) → ¬ if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩) = ∅)
58	44, 57	mp1i 13	. . . . . . . . . . . . . . . 16 ⊢ ((¬ 𝑋 ∈ 𝑈 ∧ 𝑥 = 𝑋) → ¬ if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩) = ∅)
59	58	neqned 2939	. . . . . . . . . . . . . . 15 ⊢ ((¬ 𝑋 ∈ 𝑈 ∧ 𝑥 = 𝑋) → if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩) ≠ ∅)
60	43, 59	eqnetrd 2999	. . . . . . . . . . . . . 14 ⊢ ((¬ 𝑋 ∈ 𝑈 ∧ 𝑥 = 𝑋) → if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)) ≠ ∅)
61	60	adantrl 716	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑋 ∈ 𝑈 ∧ (𝑛 = 1o ∧ 𝑥 = 𝑋)) → if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)) ≠ ∅)
62	34, 61	eqnetrrd 3000	. . . . . . . . . . . 12 ⊢ ((¬ 𝑋 ∈ 𝑈 ∧ (𝑛 = 1o ∧ 𝑥 = 𝑋)) → ⦋1o / 𝑛⦌⦋𝑋 / 𝑥⦌if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)) ≠ ∅)
63	62	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ V ∧ (¬ 𝑋 ∈ 𝑈 ∧ (𝑛 = 1o ∧ 𝑥 = 𝑋))) → ⦋1o / 𝑛⦌⦋𝑋 / 𝑥⦌if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)) ≠ ∅)
64	30, 63	eqnetrd 2999	. . . . . . . . . 10 ⊢ ((𝑋 ∈ V ∧ (¬ 𝑋 ∈ 𝑈 ∧ (𝑛 = 1o ∧ 𝑥 = 𝑋))) → (1o(𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))𝑋) ≠ ∅)
65	19, 64	eqnetrrid 3007	. . . . . . . . 9 ⊢ ((𝑋 ∈ V ∧ (¬ 𝑋 ∈ 𝑈 ∧ (𝑛 = 1o ∧ 𝑥 = 𝑋))) → ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩) ≠ ∅)
66	65	ancom2s 650	. . . . . . . 8 ⊢ ((𝑋 ∈ V ∧ ((𝑛 = 1o ∧ 𝑥 = 𝑋) ∧ ¬ 𝑋 ∈ 𝑈)) → ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩) ≠ ∅)
67	66	an12s 649	. . . . . . 7 ⊢ (((𝑛 = 1o ∧ 𝑥 = 𝑋) ∧ (𝑋 ∈ V ∧ ¬ 𝑋 ∈ 𝑈)) → ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩) ≠ ∅)
68	67	exp31 419	. . . . . 6 ⊢ (𝑛 = 1o → (𝑥 = 𝑋 → ((𝑋 ∈ V ∧ ¬ 𝑋 ∈ 𝑈) → ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩) ≠ ∅)))
69	16, 18, 68	vtoclef 3520	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑋 ∈ V ∧ ¬ 𝑋 ∈ 𝑈) → ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩) ≠ ∅))
70	7, 69	vtoclefex 37539	. . . 4 ⊢ (𝑋 ∈ V → ((𝑋 ∈ V ∧ ¬ 𝑋 ∈ 𝑈) → ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩) ≠ ∅))
71	70	anabsi5 669	. . 3 ⊢ ((𝑋 ∈ V ∧ ¬ 𝑋 ∈ 𝑈) → ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩) ≠ ∅)
72	71	necomd 2987	. 2 ⊢ ((𝑋 ∈ V ∧ ¬ 𝑋 ∈ 𝑈) → ∅ ≠ ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩))
73	72	neneqd 2937	1 ⊢ ((𝑋 ∈ V ∧ ¬ 𝑋 ∈ 𝑈) → ¬ ∅ = ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2113   ≠ wne 2932  Vcvv 3440  ⦋csb 3849  ∅c0 4285  ifcif 4479  ⟨cop 4586  ∪ cuni 4863   × cxp 5622  ‘cfv 6492  (class class class)co 7358   ∈ cmpo 7360  ωcom 7808  1^st c1st 7931  1_oc1o 8390

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1o 8397

This theorem is referenced by:  finxp1o  37597