Theorem disjprg 5105

Description: A pair collection is disjoint iff the two sets in the family have empty intersection. (Contributed by Mario Carneiro, 14-Nov-2016.)

Hypotheses

Ref	Expression
disjprg.1	⊢ (𝑥 = 𝐴 → 𝐶 = 𝐷)
disjprg.2	⊢ (𝑥 = 𝐵 → 𝐶 = 𝐸)

Assertion

Ref	Expression
disjprg	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ≠ 𝐵) → (Disj 𝑥 ∈ {𝐴, 𝐵}𝐶 ↔ (𝐷 ∩ 𝐸) = ∅))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐷   𝑥,𝐸

Allowed substitution hints:   𝐶(𝑥)   𝑉(𝑥)

Proof of Theorem disjprg

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

Step	Hyp	Ref	Expression
1		eqeq1 2767	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑦 = 𝑧 ↔ 𝐴 = 𝑧))
2		nfcv 2925	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝐴
3		nfcv 2925	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝐷
4		disjprg.1	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → 𝐶 = 𝐷)
5	2, 3, 4	csbhypf 3881	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐶 = 𝐷)
6	5	ineq1d 4172	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (⦋𝑦 / 𝑥⦌𝐶 ∩ ⦋𝑧 / 𝑥⦌𝐶) = (𝐷 ∩ ⦋𝑧 / 𝑥⦌𝐶))
7	6	eqeq1d 2765	. . . . . . 7 ⊢ (𝑦 = 𝐴 → ((⦋𝑦 / 𝑥⦌𝐶 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅ ↔ (𝐷 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅))
8	1, 7	orbi12d 931	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝑦 = 𝑧 ∨ (⦋𝑦 / 𝑥⦌𝐶 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅) ↔ (𝐴 = 𝑧 ∨ (𝐷 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅)))
9	8	ralbidv 3188	. . . . 5 ⊢ (𝑦 = 𝐴 → (∀𝑧 ∈ {𝐴, 𝐵} (𝑦 = 𝑧 ∨ (⦋𝑦 / 𝑥⦌𝐶 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅) ↔ ∀𝑧 ∈ {𝐴, 𝐵} (𝐴 = 𝑧 ∨ (𝐷 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅)))
10		eqeq1 2767	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑦 = 𝑧 ↔ 𝐵 = 𝑧))
11		nfcv 2925	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝐵
12		nfcv 2925	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝐸
13		disjprg.2	. . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → 𝐶 = 𝐸)
14	11, 12, 13	csbhypf 3881	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → ⦋𝑦 / 𝑥⦌𝐶 = 𝐸)
15	14	ineq1d 4172	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (⦋𝑦 / 𝑥⦌𝐶 ∩ ⦋𝑧 / 𝑥⦌𝐶) = (𝐸 ∩ ⦋𝑧 / 𝑥⦌𝐶))
16	15	eqeq1d 2765	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ((⦋𝑦 / 𝑥⦌𝐶 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅ ↔ (𝐸 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅))
17	10, 16	orbi12d 931	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝑦 = 𝑧 ∨ (⦋𝑦 / 𝑥⦌𝐶 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅) ↔ (𝐵 = 𝑧 ∨ (𝐸 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅)))
18	17	ralbidv 3188	. . . . 5 ⊢ (𝑦 = 𝐵 → (∀𝑧 ∈ {𝐴, 𝐵} (𝑦 = 𝑧 ∨ (⦋𝑦 / 𝑥⦌𝐶 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅) ↔ ∀𝑧 ∈ {𝐴, 𝐵} (𝐵 = 𝑧 ∨ (𝐸 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅)))
19	9, 18	ralprg 4662	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (∀𝑦 ∈ {𝐴, 𝐵}∀𝑧 ∈ {𝐴, 𝐵} (𝑦 = 𝑧 ∨ (⦋𝑦 / 𝑥⦌𝐶 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅) ↔ (∀𝑧 ∈ {𝐴, 𝐵} (𝐴 = 𝑧 ∨ (𝐷 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅) ∧ ∀𝑧 ∈ {𝐴, 𝐵} (𝐵 = 𝑧 ∨ (𝐸 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅))))
20	19	3adant3 1150	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ≠ 𝐵) → (∀𝑦 ∈ {𝐴, 𝐵}∀𝑧 ∈ {𝐴, 𝐵} (𝑦 = 𝑧 ∨ (⦋𝑦 / 𝑥⦌𝐶 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅) ↔ (∀𝑧 ∈ {𝐴, 𝐵} (𝐴 = 𝑧 ∨ (𝐷 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅) ∧ ∀𝑧 ∈ {𝐴, 𝐵} (𝐵 = 𝑧 ∨ (𝐸 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅))))
21		id 23	. . . . . . . . . 10 ⊢ (𝑧 = 𝐴 → 𝑧 = 𝐴)
22	21	eqcomd 2769	. . . . . . . . 9 ⊢ (𝑧 = 𝐴 → 𝐴 = 𝑧)
23	22	orcd 886	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → (𝐴 = 𝑧 ∨ (𝐷 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅))
24		trud 1580	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → ⊤)
25	23, 24	2thd 268	. . . . . . 7 ⊢ (𝑧 = 𝐴 → ((𝐴 = 𝑧 ∨ (𝐷 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅) ↔ ⊤))
26		eqeq2 2775	. . . . . . . 8 ⊢ (𝑧 = 𝐵 → (𝐴 = 𝑧 ↔ 𝐴 = 𝐵))
27	11, 12, 13	csbhypf 3881	. . . . . . . . . 10 ⊢ (𝑧 = 𝐵 → ⦋𝑧 / 𝑥⦌𝐶 = 𝐸)
28	27	ineq2d 4173	. . . . . . . . 9 ⊢ (𝑧 = 𝐵 → (𝐷 ∩ ⦋𝑧 / 𝑥⦌𝐶) = (𝐷 ∩ 𝐸))
29	28	eqeq1d 2765	. . . . . . . 8 ⊢ (𝑧 = 𝐵 → ((𝐷 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅ ↔ (𝐷 ∩ 𝐸) = ∅))
30	26, 29	orbi12d 931	. . . . . . 7 ⊢ (𝑧 = 𝐵 → ((𝐴 = 𝑧 ∨ (𝐷 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅) ↔ (𝐴 = 𝐵 ∨ (𝐷 ∩ 𝐸) = ∅)))
31	25, 30	ralprg 4662	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (∀𝑧 ∈ {𝐴, 𝐵} (𝐴 = 𝑧 ∨ (𝐷 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅) ↔ (⊤ ∧ (𝐴 = 𝐵 ∨ (𝐷 ∩ 𝐸) = ∅))))
32	31	3adant3 1150	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ≠ 𝐵) → (∀𝑧 ∈ {𝐴, 𝐵} (𝐴 = 𝑧 ∨ (𝐷 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅) ↔ (⊤ ∧ (𝐴 = 𝐵 ∨ (𝐷 ∩ 𝐸) = ∅))))
33		simp3 1156	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ≠ 𝐵) → 𝐴 ≠ 𝐵)
34	33	neneqd 2963	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ≠ 𝐵) → ¬ 𝐴 = 𝐵)
35		biorf 949	. . . . . . 7 ⊢ (¬ 𝐴 = 𝐵 → ((𝐷 ∩ 𝐸) = ∅ ↔ (𝐴 = 𝐵 ∨ (𝐷 ∩ 𝐸) = ∅)))
36	34, 35	syl 18	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ≠ 𝐵) → ((𝐷 ∩ 𝐸) = ∅ ↔ (𝐴 = 𝐵 ∨ (𝐷 ∩ 𝐸) = ∅)))
37		tru 1574	. . . . . . 7 ⊢ ⊤
38	37	biantrur 539	. . . . . 6 ⊢ ((𝐴 = 𝐵 ∨ (𝐷 ∩ 𝐸) = ∅) ↔ (⊤ ∧ (𝐴 = 𝐵 ∨ (𝐷 ∩ 𝐸) = ∅)))
39	36, 38	bitrdi 290	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ≠ 𝐵) → ((𝐷 ∩ 𝐸) = ∅ ↔ (⊤ ∧ (𝐴 = 𝐵 ∨ (𝐷 ∩ 𝐸) = ∅))))
40	32, 39	bitr4d 285	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ≠ 𝐵) → (∀𝑧 ∈ {𝐴, 𝐵} (𝐴 = 𝑧 ∨ (𝐷 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅) ↔ (𝐷 ∩ 𝐸) = ∅))
41		eqeq2 2775	. . . . . . . . 9 ⊢ (𝑧 = 𝐴 → (𝐵 = 𝑧 ↔ 𝐵 = 𝐴))
42		eqcom 2770	. . . . . . . . 9 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵)
43	41, 42	bitrdi 290	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → (𝐵 = 𝑧 ↔ 𝐴 = 𝐵))
44	2, 3, 4	csbhypf 3881	. . . . . . . . . . 11 ⊢ (𝑧 = 𝐴 → ⦋𝑧 / 𝑥⦌𝐶 = 𝐷)
45	44	ineq2d 4173	. . . . . . . . . 10 ⊢ (𝑧 = 𝐴 → (𝐸 ∩ ⦋𝑧 / 𝑥⦌𝐶) = (𝐸 ∩ 𝐷))
46		incom 4162	. . . . . . . . . 10 ⊢ (𝐸 ∩ 𝐷) = (𝐷 ∩ 𝐸)
47	45, 46	eqtrdi 2814	. . . . . . . . 9 ⊢ (𝑧 = 𝐴 → (𝐸 ∩ ⦋𝑧 / 𝑥⦌𝐶) = (𝐷 ∩ 𝐸))
48	47	eqeq1d 2765	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → ((𝐸 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅ ↔ (𝐷 ∩ 𝐸) = ∅))
49	43, 48	orbi12d 931	. . . . . . 7 ⊢ (𝑧 = 𝐴 → ((𝐵 = 𝑧 ∨ (𝐸 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅) ↔ (𝐴 = 𝐵 ∨ (𝐷 ∩ 𝐸) = ∅)))
50		id 23	. . . . . . . . . 10 ⊢ (𝑧 = 𝐵 → 𝑧 = 𝐵)
51	50	eqcomd 2769	. . . . . . . . 9 ⊢ (𝑧 = 𝐵 → 𝐵 = 𝑧)
52	51	orcd 886	. . . . . . . 8 ⊢ (𝑧 = 𝐵 → (𝐵 = 𝑧 ∨ (𝐸 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅))
53		trud 1580	. . . . . . . 8 ⊢ (𝑧 = 𝐵 → ⊤)
54	52, 53	2thd 268	. . . . . . 7 ⊢ (𝑧 = 𝐵 → ((𝐵 = 𝑧 ∨ (𝐸 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅) ↔ ⊤))
55	49, 54	ralprg 4662	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (∀𝑧 ∈ {𝐴, 𝐵} (𝐵 = 𝑧 ∨ (𝐸 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅) ↔ ((𝐴 = 𝐵 ∨ (𝐷 ∩ 𝐸) = ∅) ∧ ⊤)))
56	55	3adant3 1150	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ≠ 𝐵) → (∀𝑧 ∈ {𝐴, 𝐵} (𝐵 = 𝑧 ∨ (𝐸 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅) ↔ ((𝐴 = 𝐵 ∨ (𝐷 ∩ 𝐸) = ∅) ∧ ⊤)))
57	37	biantru 538	. . . . . 6 ⊢ ((𝐴 = 𝐵 ∨ (𝐷 ∩ 𝐸) = ∅) ↔ ((𝐴 = 𝐵 ∨ (𝐷 ∩ 𝐸) = ∅) ∧ ⊤))
58	36, 57	bitrdi 290	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ≠ 𝐵) → ((𝐷 ∩ 𝐸) = ∅ ↔ ((𝐴 = 𝐵 ∨ (𝐷 ∩ 𝐸) = ∅) ∧ ⊤)))
59	56, 58	bitr4d 285	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ≠ 𝐵) → (∀𝑧 ∈ {𝐴, 𝐵} (𝐵 = 𝑧 ∨ (𝐸 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅) ↔ (𝐷 ∩ 𝐸) = ∅))
60	40, 59	anbi12d 643	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ≠ 𝐵) → ((∀𝑧 ∈ {𝐴, 𝐵} (𝐴 = 𝑧 ∨ (𝐷 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅) ∧ ∀𝑧 ∈ {𝐴, 𝐵} (𝐵 = 𝑧 ∨ (𝐸 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅)) ↔ ((𝐷 ∩ 𝐸) = ∅ ∧ (𝐷 ∩ 𝐸) = ∅)))
61	20, 60	bitrd 282	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ≠ 𝐵) → (∀𝑦 ∈ {𝐴, 𝐵}∀𝑧 ∈ {𝐴, 𝐵} (𝑦 = 𝑧 ∨ (⦋𝑦 / 𝑥⦌𝐶 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅) ↔ ((𝐷 ∩ 𝐸) = ∅ ∧ (𝐷 ∩ 𝐸) = ∅)))
62		disjors 5092	. 2 ⊢ (Disj 𝑥 ∈ {𝐴, 𝐵}𝐶 ↔ ∀𝑦 ∈ {𝐴, 𝐵}∀𝑧 ∈ {𝐴, 𝐵} (𝑦 = 𝑧 ∨ (⦋𝑦 / 𝑥⦌𝐶 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅))
63		pm4.24 573	. 2 ⊢ ((𝐷 ∩ 𝐸) = ∅ ↔ ((𝐷 ∩ 𝐸) = ∅ ∧ (𝐷 ∩ 𝐸) = ∅))
64	61, 62, 63	3bitr4g 317	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ≠ 𝐵) → (Disj 𝑥 ∈ {𝐴, 𝐵}𝐶 ↔ (𝐷 ∩ 𝐸) = ∅))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∧ w3a 1103   = wceq 1570  ⊤wtru 1571   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  ⦋csb 3853   ∩ cin 3904  ∅c0 4286  {cpr 4591  Disj wdisj 5076

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-nul 4287  df-sn 4590  df-pr 4592  df-disj 5077

This theorem is used by:  disjdifprg  32929  unelldsys  34557  pmeasmono  34723  probun  34818  meadjun  47204