Theorem disjxun 5101

Description: The union of two disjoint collections. (Contributed by Mario Carneiro, 14-Nov-2016.)

Hypothesis

Ref	Expression
disjxun.1	⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷)

Assertion

Ref	Expression
disjxun	⊢ ((𝐴 ∩ 𝐵) = ∅ → (Disj 𝑥 ∈ (𝐴 ∪ 𝐵)𝐶 ↔ (Disj 𝑥 ∈ 𝐴 𝐶 ∧ Disj 𝑥 ∈ 𝐵 𝐶 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = ∅)))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑦,𝐶   𝑥,𝐷

Allowed substitution hints:   𝐶(𝑥)   𝐷(𝑦)

Proof of Theorem disjxun

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		disjel 4410	. . . . . . . . . . 11 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 ∈ 𝐵)
2		eleq1w 2844	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
3	2	notbid 321	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ 𝐵 ↔ ¬ 𝑦 ∈ 𝐵))
4	1, 3	syl5ibcom 248	. . . . . . . . . 10 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝑥 ∈ 𝐴) → (𝑥 = 𝑦 → ¬ 𝑦 ∈ 𝐵))
5	4	con2d 135	. . . . . . . . 9 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ 𝐵 → ¬ 𝑥 = 𝑦))
6	5	impr 460	. . . . . . . 8 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ¬ 𝑥 = 𝑦)
7		biorf 950	. . . . . . . 8 ⊢ (¬ 𝑥 = 𝑦 → ((𝐶 ∩ 𝐷) = ∅ ↔ (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅)))
8	6, 7	syl 18	. . . . . . 7 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ((𝐶 ∩ 𝐷) = ∅ ↔ (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅)))
9	8	bicomd 226	. . . . . 6 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ((𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅) ↔ (𝐶 ∩ 𝐷) = ∅))
10	9	2ralbidva 3225	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = ∅))
11	10	anbi2d 642	. . . 4 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅)) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = ∅)))
12		ralunb 4143	. . . . . 6 ⊢ (∀𝑦 ∈ (𝐴 ∪ 𝐵)(𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅) ↔ (∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅) ∧ ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅)))
13	12	ralbii 3109	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ (𝐴 ∪ 𝐵)(𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅) ↔ ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅) ∧ ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅)))
14		nfv 1947	. . . . . 6 ⊢ Ⅎ𝑧∀𝑦 ∈ (𝐴 ∪ 𝐵)(𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅)
15		nfcv 2923	. . . . . . 7 ⊢ Ⅎ𝑥(𝐴 ∪ 𝐵)
16		nfv 1947	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑧 = 𝑤
17		nfcsb1v 3871	. . . . . . . . . 10 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐶
18		nfcsb1v 3871	. . . . . . . . . 10 ⊢ Ⅎ𝑥⦋𝑤 / 𝑥⦌𝐶
19	17, 18	nfin 4170	. . . . . . . . 9 ⊢ Ⅎ𝑥(⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶)
20	19	nfeq1 2938	. . . . . . . 8 ⊢ Ⅎ𝑥(⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅
21	16, 20	nfor 1937	. . . . . . 7 ⊢ Ⅎ𝑥(𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅)
22	15, 21	nfralw 3310	. . . . . 6 ⊢ Ⅎ𝑥∀𝑤 ∈ (𝐴 ∪ 𝐵)(𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅)
23		equequ2 2059	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝑥 = 𝑤 ↔ 𝑥 = 𝑦))
24		nfcv 2923	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝑦
25		nfcv 2923	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝐷
26		disjxun.1	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷)
27	24, 25, 26	csbhypf 3875	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑦 → ⦋𝑤 / 𝑥⦌𝐶 = 𝐷)
28	27	ineq2d 4166	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = (𝐶 ∩ 𝐷))
29	28	eqeq1d 2763	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → ((𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅ ↔ (𝐶 ∩ 𝐷) = ∅))
30	23, 29	orbi12d 932	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → ((𝑥 = 𝑤 ∨ (𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅) ↔ (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅)))
31	30	cbvralvw 3241	. . . . . . 7 ⊢ (∀𝑤 ∈ (𝐴 ∪ 𝐵)(𝑥 = 𝑤 ∨ (𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅) ↔ ∀𝑦 ∈ (𝐴 ∪ 𝐵)(𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅))
32		equequ1 2058	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑤 ↔ 𝑧 = 𝑤))
33		csbeq1a 3861	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → 𝐶 = ⦋𝑧 / 𝑥⦌𝐶)
34	33	ineq1d 4165	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶))
35	34	eqeq1d 2763	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → ((𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅ ↔ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅))
36	32, 35	orbi12d 932	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → ((𝑥 = 𝑤 ∨ (𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅) ↔ (𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅)))
37	36	ralbidv 3186	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (∀𝑤 ∈ (𝐴 ∪ 𝐵)(𝑥 = 𝑤 ∨ (𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅) ↔ ∀𝑤 ∈ (𝐴 ∪ 𝐵)(𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅)))
38	31, 37	bitr3id 288	. . . . . 6 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ (𝐴 ∪ 𝐵)(𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅) ↔ ∀𝑤 ∈ (𝐴 ∪ 𝐵)(𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅)))
39	14, 22, 38	cbvralw 3305	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ (𝐴 ∪ 𝐵)(𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅) ↔ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ (𝐴 ∪ 𝐵)(𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅))
40		r19.26 3123	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅) ∧ ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅)) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅)))
41	13, 39, 40	3bitr3i 304	. . . 4 ⊢ (∀𝑧 ∈ 𝐴 ∀𝑤 ∈ (𝐴 ∪ 𝐵)(𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅)))
42	26	disjor 5085	. . . . 5 ⊢ (Disj 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅))
43	42	anbi1i 636	. . . 4 ⊢ ((Disj 𝑥 ∈ 𝐴 𝐶 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = ∅) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = ∅))
44	11, 41, 43	3bitr4g 317	. . 3 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (∀𝑧 ∈ 𝐴 ∀𝑤 ∈ (𝐴 ∪ 𝐵)(𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅) ↔ (Disj 𝑥 ∈ 𝐴 𝐶 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = ∅)))
45		nfv 1947	. . . . . . . . . 10 ⊢ Ⅎ𝑤(𝑧 = 𝑥 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ 𝐶) = ∅)
46		equequ2 2059	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → (𝑧 = 𝑥 ↔ 𝑧 = 𝑤))
47		csbeq1a 3861	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → 𝐶 = ⦋𝑤 / 𝑥⦌𝐶)
48	47	ineq2d 4166	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → (⦋𝑧 / 𝑥⦌𝐶 ∩ 𝐶) = (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶))
49	48	eqeq1d 2763	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → ((⦋𝑧 / 𝑥⦌𝐶 ∩ 𝐶) = ∅ ↔ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅))
50	46, 49	orbi12d 932	. . . . . . . . . 10 ⊢ (𝑥 = 𝑤 → ((𝑧 = 𝑥 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ 𝐶) = ∅) ↔ (𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅)))
51	45, 21, 50	cbvralw 3305	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 (𝑧 = 𝑥 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ 𝐶) = ∅) ↔ ∀𝑤 ∈ 𝐴 (𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅))
52		equequ1 2058	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑦 → (𝑧 = 𝑥 ↔ 𝑦 = 𝑥))
53		equcom 2051	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
54	52, 53	bitrdi 290	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → (𝑧 = 𝑥 ↔ 𝑥 = 𝑦))
55	24, 25, 26	csbhypf 3875	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑦 → ⦋𝑧 / 𝑥⦌𝐶 = 𝐷)
56	55	ineq1d 4165	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑦 → (⦋𝑧 / 𝑥⦌𝐶 ∩ 𝐶) = (𝐷 ∩ 𝐶))
57		incom 4155	. . . . . . . . . . . . 13 ⊢ (𝐷 ∩ 𝐶) = (𝐶 ∩ 𝐷)
58	56, 57	eqtrdi 2812	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑦 → (⦋𝑧 / 𝑥⦌𝐶 ∩ 𝐶) = (𝐶 ∩ 𝐷))
59	58	eqeq1d 2763	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → ((⦋𝑧 / 𝑥⦌𝐶 ∩ 𝐶) = ∅ ↔ (𝐶 ∩ 𝐷) = ∅))
60	54, 59	orbi12d 932	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → ((𝑧 = 𝑥 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ 𝐶) = ∅) ↔ (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅)))
61	60	ralbidv 3186	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → (∀𝑥 ∈ 𝐴 (𝑧 = 𝑥 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ 𝐶) = ∅) ↔ ∀𝑥 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅)))
62	51, 61	bitr3id 288	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (∀𝑤 ∈ 𝐴 (𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅) ↔ ∀𝑥 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅)))
63	62	cbvralvw 3241	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐴 (𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅) ↔ ∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅))
64		ralcom 3291	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅))
65	63, 64	bitri 278	. . . . . 6 ⊢ (∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐴 (𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅))
66	65, 10	bitrid 286	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐴 (𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = ∅))
67	66	anbi1d 643	. . . 4 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐴 (𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅) ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅)) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = ∅ ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅))))
68		ralunb 4143	. . . . . 6 ⊢ (∀𝑤 ∈ (𝐴 ∪ 𝐵)(𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅) ↔ (∀𝑤 ∈ 𝐴 (𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅) ∧ ∀𝑤 ∈ 𝐵 (𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅)))
69	68	ralbii 3109	. . . . 5 ⊢ (∀𝑧 ∈ 𝐵 ∀𝑤 ∈ (𝐴 ∪ 𝐵)(𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅) ↔ ∀𝑧 ∈ 𝐵 (∀𝑤 ∈ 𝐴 (𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅) ∧ ∀𝑤 ∈ 𝐵 (𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅)))
70		r19.26 3123	. . . . 5 ⊢ (∀𝑧 ∈ 𝐵 (∀𝑤 ∈ 𝐴 (𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅) ∧ ∀𝑤 ∈ 𝐵 (𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅)) ↔ (∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐴 (𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅) ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅)))
71	69, 70	bitri 278	. . . 4 ⊢ (∀𝑧 ∈ 𝐵 ∀𝑤 ∈ (𝐴 ∪ 𝐵)(𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅) ↔ (∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐴 (𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅) ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅)))
72		disjors 5086	. . . . 5 ⊢ (Disj 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅))
73	72	anbi2ci 637	. . . 4 ⊢ ((Disj 𝑥 ∈ 𝐵 𝐶 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = ∅) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = ∅ ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅)))
74	67, 71, 73	3bitr4g 317	. . 3 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (∀𝑧 ∈ 𝐵 ∀𝑤 ∈ (𝐴 ∪ 𝐵)(𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅) ↔ (Disj 𝑥 ∈ 𝐵 𝐶 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = ∅)))
75	44, 74	anbi12d 644	. 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((∀𝑧 ∈ 𝐴 ∀𝑤 ∈ (𝐴 ∪ 𝐵)(𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅) ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ (𝐴 ∪ 𝐵)(𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅)) ↔ ((Disj 𝑥 ∈ 𝐴 𝐶 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = ∅) ∧ (Disj 𝑥 ∈ 𝐵 𝐶 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = ∅))))
76		disjors 5086	. . 3 ⊢ (Disj 𝑥 ∈ (𝐴 ∪ 𝐵)𝐶 ↔ ∀𝑧 ∈ (𝐴 ∪ 𝐵)∀𝑤 ∈ (𝐴 ∪ 𝐵)(𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅))
77		ralunb 4143	. . 3 ⊢ (∀𝑧 ∈ (𝐴 ∪ 𝐵)∀𝑤 ∈ (𝐴 ∪ 𝐵)(𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅) ↔ (∀𝑧 ∈ 𝐴 ∀𝑤 ∈ (𝐴 ∪ 𝐵)(𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅) ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ (𝐴 ∪ 𝐵)(𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅)))
78	76, 77	bitri 278	. 2 ⊢ (Disj 𝑥 ∈ (𝐴 ∪ 𝐵)𝐶 ↔ (∀𝑧 ∈ 𝐴 ∀𝑤 ∈ (𝐴 ∪ 𝐵)(𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅) ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ (𝐴 ∪ 𝐵)(𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅)))
79		df-3an 1105	. . 3 ⊢ ((Disj 𝑥 ∈ 𝐴 𝐶 ∧ Disj 𝑥 ∈ 𝐵 𝐶 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = ∅) ↔ ((Disj 𝑥 ∈ 𝐴 𝐶 ∧ Disj 𝑥 ∈ 𝐵 𝐶) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = ∅))
80		anandir 690	. . 3 ⊢ (((Disj 𝑥 ∈ 𝐴 𝐶 ∧ Disj 𝑥 ∈ 𝐵 𝐶) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = ∅) ↔ ((Disj 𝑥 ∈ 𝐴 𝐶 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = ∅) ∧ (Disj 𝑥 ∈ 𝐵 𝐶 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = ∅)))
81	79, 80	bitri 278	. 2 ⊢ ((Disj 𝑥 ∈ 𝐴 𝐶 ∧ Disj 𝑥 ∈ 𝐵 𝐶 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = ∅) ↔ ((Disj 𝑥 ∈ 𝐴 𝐶 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = ∅) ∧ (Disj 𝑥 ∈ 𝐵 𝐶 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = ∅)))
82	75, 78, 81	3bitr4g 317	1 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (Disj 𝑥 ∈ (𝐴 ∪ 𝐵)𝐶 ↔ (Disj 𝑥 ∈ 𝐴 𝐶 ∧ Disj 𝑥 ∈ 𝐵 𝐶 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = ∅)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ⦋csb 3847   ∪ cun 3897   ∩ cin 3898  ∅c0 4279  Disj wdisj 5070

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3078  df-rmo 3366  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-nul 4280  df-disj 5071

This theorem is used by: (None)