Theorem finxpreclem2 36574

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 7492	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))
3		nfcv 2901	. . . . . . . 8 ⊢ Ⅎ𝑥⟨1o, 𝑋⟩
4	2, 3	nffv 6900	. . . . . . 7 ⊢ Ⅎ𝑥((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩)
5		nfcv 2901	. . . . . . 7 ⊢ Ⅎ𝑥∅
6	4, 5	nfne 3041	. . . . . 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 7491	. . . . . . . . . 10 ⊢ Ⅎ𝑛(𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))
11		nfcv 2901	. . . . . . . . . 10 ⊢ Ⅎ𝑛⟨1o, 𝑋⟩
12	10, 11	nffv 6900	. . . . . . . . 9 ⊢ Ⅎ𝑛((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩)
13		nfcv 2901	. . . . . . . . 9 ⊢ Ⅎ𝑛∅
14	12, 13	nfne 3041	. . . . . . . 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 8641	. . . . . . 7 ⊢ 1o ∈ ω
18	17	elexi 3492	. . . . . 6 ⊢ 1o ∈ V
19		df-ov 7414	. . . . . . . . . 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 5306	. . . . . . . . . . . . . . . 16 ⊢ ∅ ∈ V
21		opex 5463	. . . . . . . . . . . . . . . . 17 ⊢ ⟨∪ 𝑛, (1st ‘𝑥)⟩ ∈ V
22		opex 5463	. . . . . . . . . . . . . . . . 17 ⊢ ⟨𝑛, 𝑥⟩ ∈ V
23	21, 22	ifex 4577	. . . . . . . . . . . . . . . 16 ⊢ if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩) ∈ V
24	20, 23	ifex 4577	. . . . . . . . . . . . . . 15 ⊢ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)) ∈ V
25	24	csbex 5310	. . . . . . . . . . . . . 14 ⊢ ⦋𝑋 / 𝑥⦌if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)) ∈ V
26	25	csbex 5310	. . . . . . . . . . . . 13 ⊢ ⦋1o / 𝑛⦌⦋𝑋 / 𝑥⦌if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)) ∈ V
27		eqid 2730	. . . . . . . . . . . . . 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 7558	. . . . . . . . . . . . 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 1450	. . . . . . . . . . . 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 479	. . . . . . . . . . 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 3906	. . . . . . . . . . . . . . 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 3906	. . . . . . . . . . . . . . 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 2792	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . . 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 2819	. . . . . . . . . . . . . . . . . . . 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		pm3.14 992	. . . . . . . . . . . . . . . . . . 19 ⊢ ((¬ 𝑛 = 1o ∨ ¬ 𝑥 ∈ 𝑈) → ¬ (𝑛 = 1o ∧ 𝑥 ∈ 𝑈))
39	38	olcs 872	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ 𝑥 ∈ 𝑈 → ¬ (𝑛 = 1o ∧ 𝑥 ∈ 𝑈))
40	37, 39	syl6 35	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑋 ∈ 𝑈 → (𝑥 = 𝑋 → ¬ (𝑛 = 1o ∧ 𝑥 ∈ 𝑈)))
41		iffalse 4536	. . . . . . . . . . . . . . . . 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 405	. . . . . . . . . . . . . . 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 4578	. . . . . . . . . . . . . . . . 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 7731	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ∪ 𝑛 ∈ V
46		fvex 6903	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1st ‘𝑥) ∈ V
47	45, 46	opnzi 5473	. . . . . . . . . . . . . . . . . . . 20 ⊢ ⟨∪ 𝑛, (1st ‘𝑥)⟩ ≠ ∅
48	47	neii 2940	. . . . . . . . . . . . . . . . . . 19 ⊢ ¬ ⟨∪ 𝑛, (1st ‘𝑥)⟩ = ∅
49		eqeq1 2734	. . . . . . . . . . . . . . . . . . 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 3476	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑛 ∈ V
52		vex 3476	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑥 ∈ V
53	51, 52	opnzi 5473	. . . . . . . . . . . . . . . . . . . 20 ⊢ ⟨𝑛, 𝑥⟩ ≠ ∅
54	53	neii 2940	. . . . . . . . . . . . . . . . . . 19 ⊢ ¬ ⟨𝑛, 𝑥⟩ = ∅
55		eqeq1 2734	. . . . . . . . . . . . . . . . . . 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 853	. . . . . . . . . . . . . . . . 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 2945	. . . . . . . . . . . . . . 15 ⊢ ((¬ 𝑋 ∈ 𝑈 ∧ 𝑥 = 𝑋) → if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩) ≠ ∅)
60	43, 59	eqnetrd 3006	. . . . . . . . . . . . . 14 ⊢ ((¬ 𝑋 ∈ 𝑈 ∧ 𝑥 = 𝑋) → if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)) ≠ ∅)
61	60	adantrl 712	. . . . . . . . . . . . 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 3007	. . . . . . . . . . . 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 480	. . . . . . . . . . 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 3006	. . . . . . . . . 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 3014	. . . . . . . . 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 646	. . . . . . . 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 645	. . . . . . 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 418	. . . . . 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 3549	. . . . 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 36518	. . . 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 665	. . 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 2994	. 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 2943	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 394   ∨ wo 843   = wceq 1539   ∈ wcel 2104   ≠ wne 2938  Vcvv 3472  ⦋csb 3892  ∅c0 4321  ifcif 4527  ⟨cop 4633  ∪ cuni 4907   × cxp 5673  ‘cfv 6542  (class class class)co 7411   ∈ cmpo 7413  ωcom 7857  1^st c1st 7975  1_oc1o 8461

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1o 8468

This theorem is referenced by:  finxp1o  36576