Theorem finxpreclem2 36575

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 1916	. . . . . 6 ⊢ Ⅎ𝑥(𝑋 ∈ V ∧ ¬ 𝑋 ∈ 𝑈)
2		nfmpo2 7493	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))
3		nfcv 2902	. . . . . . . 8 ⊢ Ⅎ𝑥⟨1o, 𝑋⟩
4	2, 3	nffv 6902	. . . . . . 7 ⊢ Ⅎ𝑥((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩)
5		nfcv 2902	. . . . . . 7 ⊢ Ⅎ𝑥∅
6	4, 5	nfne 3042	. . . . . 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 1898	. . . . 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 1916	. . . . . . 7 ⊢ Ⅎ𝑛 𝑥 = 𝑋
9		nfv 1916	. . . . . . . 8 ⊢ Ⅎ𝑛(𝑋 ∈ V ∧ ¬ 𝑋 ∈ 𝑈)
10		nfmpo1 7492	. . . . . . . . . 10 ⊢ Ⅎ𝑛(𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))
11		nfcv 2902	. . . . . . . . . 10 ⊢ Ⅎ𝑛⟨1o, 𝑋⟩
12	10, 11	nffv 6902	. . . . . . . . 9 ⊢ Ⅎ𝑛((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩)
13		nfcv 2902	. . . . . . . . 9 ⊢ Ⅎ𝑛∅
14	12, 13	nfne 3042	. . . . . . . 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 1898	. . . . . . 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 1898	. . . . . 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 8642	. . . . . . 7 ⊢ 1o ∈ ω
18	17	elexi 3493	. . . . . 6 ⊢ 1o ∈ V
19		df-ov 7415	. . . . . . . . . 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 5308	. . . . . . . . . . . . . . . 16 ⊢ ∅ ∈ V
21		opex 5465	. . . . . . . . . . . . . . . . 17 ⊢ ⟨∪ 𝑛, (1st ‘𝑥)⟩ ∈ V
22		opex 5465	. . . . . . . . . . . . . . . . 17 ⊢ ⟨𝑛, 𝑥⟩ ∈ V
23	21, 22	ifex 4579	. . . . . . . . . . . . . . . 16 ⊢ if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩) ∈ V
24	20, 23	ifex 4579	. . . . . . . . . . . . . . 15 ⊢ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)) ∈ V
25	24	csbex 5312	. . . . . . . . . . . . . 14 ⊢ ⦋𝑋 / 𝑥⦌if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)) ∈ V
26	25	csbex 5312	. . . . . . . . . . . . 13 ⊢ ⦋1o / 𝑛⦌⦋𝑋 / 𝑥⦌if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)) ∈ V
27		eqid 2731	. . . . . . . . . . . . . 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 7559	. . . . . . . . . . . . 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 1451	. . . . . . . . . . . 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 3908	. . . . . . . . . . . . . . 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 3908	. . . . . . . . . . . . . . 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 2820	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝑈 ↔ 𝑋 ∈ 𝑈))
36	35	notbid 317	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑋 → (¬ 𝑥 ∈ 𝑈 ↔ ¬ 𝑋 ∈ 𝑈))
37	36	biimprcd 249	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ 𝑋 ∈ 𝑈 → (𝑥 = 𝑋 → ¬ 𝑥 ∈ 𝑈))
38		pm3.14 993	. . . . . . . . . . . . . . . . . . 19 ⊢ ((¬ 𝑛 = 1o ∨ ¬ 𝑥 ∈ 𝑈) → ¬ (𝑛 = 1o ∧ 𝑥 ∈ 𝑈))
39	38	olcs 873	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ 𝑥 ∈ 𝑈 → ¬ (𝑛 = 1o ∧ 𝑥 ∈ 𝑈))
40	37, 39	syl6 35	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑋 ∈ 𝑈 → (𝑥 = 𝑋 → ¬ (𝑛 = 1o ∧ 𝑥 ∈ 𝑈)))
41		iffalse 4538	. . . . . . . . . . . . . . . . 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 4580	. . . . . . . . . . . . . . . . 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 7732	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ∪ 𝑛 ∈ V
46		fvex 6905	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1st ‘𝑥) ∈ V
47	45, 46	opnzi 5475	. . . . . . . . . . . . . . . . . . . 20 ⊢ ⟨∪ 𝑛, (1st ‘𝑥)⟩ ≠ ∅
48	47	neii 2941	. . . . . . . . . . . . . . . . . . 19 ⊢ ¬ ⟨∪ 𝑛, (1st ‘𝑥)⟩ = ∅
49		eqeq1 2735	. . . . . . . . . . . . . . . . . . 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 326	. . . . . . . . . . . . . . . . . 18 ⊢ (if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩) = ⟨∪ 𝑛, (1st ‘𝑥)⟩ → ¬ if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩) = ∅)
51		vex 3477	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑛 ∈ V
52		vex 3477	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑥 ∈ V
53	51, 52	opnzi 5475	. . . . . . . . . . . . . . . . . . . 20 ⊢ ⟨𝑛, 𝑥⟩ ≠ ∅
54	53	neii 2941	. . . . . . . . . . . . . . . . . . 19 ⊢ ¬ ⟨𝑛, 𝑥⟩ = ∅
55		eqeq1 2735	. . . . . . . . . . . . . . . . . . 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 326	. . . . . . . . . . . . . . . . . 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 854	. . . . . . . . . . . . . . . . 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 2946	. . . . . . . . . . . . . . 15 ⊢ ((¬ 𝑋 ∈ 𝑈 ∧ 𝑥 = 𝑋) → if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩) ≠ ∅)
60	43, 59	eqnetrd 3007	. . . . . . . . . . . . . 14 ⊢ ((¬ 𝑋 ∈ 𝑈 ∧ 𝑥 = 𝑋) → if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)) ≠ ∅)
61	60	adantrl 713	. . . . . . . . . . . . 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 3008	. . . . . . . . . . . 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 3007	. . . . . . . . . 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 3015	. . . . . . . . 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 647	. . . . . . . 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 646	. . . . . . 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 3547	. . . . 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 36519	. . . 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 666	. . 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 2995	. 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 2944	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 844   = wceq 1540   ∈ wcel 2105   ≠ wne 2939  Vcvv 3473  ⦋csb 3894  ∅c0 4323  ifcif 4529  ⟨cop 4635  ∪ cuni 4909   × cxp 5675  ‘cfv 6544  (class class class)co 7412   ∈ cmpo 7414  ωcom 7858  1^st c1st 7976  1_oc1o 8462

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7728

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  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 7415  df-oprab 7416  df-mpo 7417  df-om 7859  df-1o 8469

This theorem is referenced by:  finxp1o  36577