Theorem disjprg 5145

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 2737	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑦 = 𝑧 ↔ 𝐴 = 𝑧))
2		nfcv 2904	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝐴
3		nfcv 2904	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝐷
4		disjprg.1	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → 𝐶 = 𝐷)
5	2, 3, 4	csbhypf 3923	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐶 = 𝐷)
6	5	ineq1d 4212	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (⦋𝑦 / 𝑥⦌𝐶 ∩ ⦋𝑧 / 𝑥⦌𝐶) = (𝐷 ∩ ⦋𝑧 / 𝑥⦌𝐶))
7	6	eqeq1d 2735	. . . . . . 7 ⊢ (𝑦 = 𝐴 → ((⦋𝑦 / 𝑥⦌𝐶 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅ ↔ (𝐷 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅))
8	1, 7	orbi12d 918	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝑦 = 𝑧 ∨ (⦋𝑦 / 𝑥⦌𝐶 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅) ↔ (𝐴 = 𝑧 ∨ (𝐷 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅)))
9	8	ralbidv 3178	. . . . 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 2737	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑦 = 𝑧 ↔ 𝐵 = 𝑧))
11		nfcv 2904	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝐵
12		nfcv 2904	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝐸
13		disjprg.2	. . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → 𝐶 = 𝐸)
14	11, 12, 13	csbhypf 3923	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → ⦋𝑦 / 𝑥⦌𝐶 = 𝐸)
15	14	ineq1d 4212	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (⦋𝑦 / 𝑥⦌𝐶 ∩ ⦋𝑧 / 𝑥⦌𝐶) = (𝐸 ∩ ⦋𝑧 / 𝑥⦌𝐶))
16	15	eqeq1d 2735	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ((⦋𝑦 / 𝑥⦌𝐶 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅ ↔ (𝐸 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅))
17	10, 16	orbi12d 918	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝑦 = 𝑧 ∨ (⦋𝑦 / 𝑥⦌𝐶 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅) ↔ (𝐵 = 𝑧 ∨ (𝐸 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅)))
18	17	ralbidv 3178	. . . . 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 4699	. . . 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 1133	. . 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 22	. . . . . . . . . 10 ⊢ (𝑧 = 𝐴 → 𝑧 = 𝐴)
22	21	eqcomd 2739	. . . . . . . . 9 ⊢ (𝑧 = 𝐴 → 𝐴 = 𝑧)
23	22	orcd 872	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → (𝐴 = 𝑧 ∨ (𝐷 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅))
24		trud 1552	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → ⊤)
25	23, 24	2thd 265	. . . . . . 7 ⊢ (𝑧 = 𝐴 → ((𝐴 = 𝑧 ∨ (𝐷 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅) ↔ ⊤))
26		eqeq2 2745	. . . . . . . 8 ⊢ (𝑧 = 𝐵 → (𝐴 = 𝑧 ↔ 𝐴 = 𝐵))
27	11, 12, 13	csbhypf 3923	. . . . . . . . . 10 ⊢ (𝑧 = 𝐵 → ⦋𝑧 / 𝑥⦌𝐶 = 𝐸)
28	27	ineq2d 4213	. . . . . . . . 9 ⊢ (𝑧 = 𝐵 → (𝐷 ∩ ⦋𝑧 / 𝑥⦌𝐶) = (𝐷 ∩ 𝐸))
29	28	eqeq1d 2735	. . . . . . . 8 ⊢ (𝑧 = 𝐵 → ((𝐷 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅ ↔ (𝐷 ∩ 𝐸) = ∅))
30	26, 29	orbi12d 918	. . . . . . 7 ⊢ (𝑧 = 𝐵 → ((𝐴 = 𝑧 ∨ (𝐷 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅) ↔ (𝐴 = 𝐵 ∨ (𝐷 ∩ 𝐸) = ∅)))
31	25, 30	ralprg 4699	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (∀𝑧 ∈ {𝐴, 𝐵} (𝐴 = 𝑧 ∨ (𝐷 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅) ↔ (⊤ ∧ (𝐴 = 𝐵 ∨ (𝐷 ∩ 𝐸) = ∅))))
32	31	3adant3 1133	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ≠ 𝐵) → (∀𝑧 ∈ {𝐴, 𝐵} (𝐴 = 𝑧 ∨ (𝐷 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅) ↔ (⊤ ∧ (𝐴 = 𝐵 ∨ (𝐷 ∩ 𝐸) = ∅))))
33		simp3 1139	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ≠ 𝐵) → 𝐴 ≠ 𝐵)
34	33	neneqd 2946	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ≠ 𝐵) → ¬ 𝐴 = 𝐵)
35		biorf 936	. . . . . . 7 ⊢ (¬ 𝐴 = 𝐵 → ((𝐷 ∩ 𝐸) = ∅ ↔ (𝐴 = 𝐵 ∨ (𝐷 ∩ 𝐸) = ∅)))
36	34, 35	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ≠ 𝐵) → ((𝐷 ∩ 𝐸) = ∅ ↔ (𝐴 = 𝐵 ∨ (𝐷 ∩ 𝐸) = ∅)))
37		tru 1546	. . . . . . 7 ⊢ ⊤
38	37	biantrur 532	. . . . . 6 ⊢ ((𝐴 = 𝐵 ∨ (𝐷 ∩ 𝐸) = ∅) ↔ (⊤ ∧ (𝐴 = 𝐵 ∨ (𝐷 ∩ 𝐸) = ∅)))
39	36, 38	bitrdi 287	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ≠ 𝐵) → ((𝐷 ∩ 𝐸) = ∅ ↔ (⊤ ∧ (𝐴 = 𝐵 ∨ (𝐷 ∩ 𝐸) = ∅))))
40	32, 39	bitr4d 282	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ≠ 𝐵) → (∀𝑧 ∈ {𝐴, 𝐵} (𝐴 = 𝑧 ∨ (𝐷 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅) ↔ (𝐷 ∩ 𝐸) = ∅))
41		eqeq2 2745	. . . . . . . . 9 ⊢ (𝑧 = 𝐴 → (𝐵 = 𝑧 ↔ 𝐵 = 𝐴))
42		eqcom 2740	. . . . . . . . 9 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵)
43	41, 42	bitrdi 287	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → (𝐵 = 𝑧 ↔ 𝐴 = 𝐵))
44	2, 3, 4	csbhypf 3923	. . . . . . . . . . 11 ⊢ (𝑧 = 𝐴 → ⦋𝑧 / 𝑥⦌𝐶 = 𝐷)
45	44	ineq2d 4213	. . . . . . . . . 10 ⊢ (𝑧 = 𝐴 → (𝐸 ∩ ⦋𝑧 / 𝑥⦌𝐶) = (𝐸 ∩ 𝐷))
46		incom 4202	. . . . . . . . . 10 ⊢ (𝐸 ∩ 𝐷) = (𝐷 ∩ 𝐸)
47	45, 46	eqtrdi 2789	. . . . . . . . 9 ⊢ (𝑧 = 𝐴 → (𝐸 ∩ ⦋𝑧 / 𝑥⦌𝐶) = (𝐷 ∩ 𝐸))
48	47	eqeq1d 2735	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → ((𝐸 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅ ↔ (𝐷 ∩ 𝐸) = ∅))
49	43, 48	orbi12d 918	. . . . . . 7 ⊢ (𝑧 = 𝐴 → ((𝐵 = 𝑧 ∨ (𝐸 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅) ↔ (𝐴 = 𝐵 ∨ (𝐷 ∩ 𝐸) = ∅)))
50		id 22	. . . . . . . . . 10 ⊢ (𝑧 = 𝐵 → 𝑧 = 𝐵)
51	50	eqcomd 2739	. . . . . . . . 9 ⊢ (𝑧 = 𝐵 → 𝐵 = 𝑧)
52	51	orcd 872	. . . . . . . 8 ⊢ (𝑧 = 𝐵 → (𝐵 = 𝑧 ∨ (𝐸 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅))
53		trud 1552	. . . . . . . 8 ⊢ (𝑧 = 𝐵 → ⊤)
54	52, 53	2thd 265	. . . . . . 7 ⊢ (𝑧 = 𝐵 → ((𝐵 = 𝑧 ∨ (𝐸 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅) ↔ ⊤))
55	49, 54	ralprg 4699	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (∀𝑧 ∈ {𝐴, 𝐵} (𝐵 = 𝑧 ∨ (𝐸 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅) ↔ ((𝐴 = 𝐵 ∨ (𝐷 ∩ 𝐸) = ∅) ∧ ⊤)))
56	55	3adant3 1133	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ≠ 𝐵) → (∀𝑧 ∈ {𝐴, 𝐵} (𝐵 = 𝑧 ∨ (𝐸 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅) ↔ ((𝐴 = 𝐵 ∨ (𝐷 ∩ 𝐸) = ∅) ∧ ⊤)))
57	37	biantru 531	. . . . . 6 ⊢ ((𝐴 = 𝐵 ∨ (𝐷 ∩ 𝐸) = ∅) ↔ ((𝐴 = 𝐵 ∨ (𝐷 ∩ 𝐸) = ∅) ∧ ⊤))
58	36, 57	bitrdi 287	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ≠ 𝐵) → ((𝐷 ∩ 𝐸) = ∅ ↔ ((𝐴 = 𝐵 ∨ (𝐷 ∩ 𝐸) = ∅) ∧ ⊤)))
59	56, 58	bitr4d 282	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ≠ 𝐵) → (∀𝑧 ∈ {𝐴, 𝐵} (𝐵 = 𝑧 ∨ (𝐸 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅) ↔ (𝐷 ∩ 𝐸) = ∅))
60	40, 59	anbi12d 632	. . 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 279	. 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 5130	. 2 ⊢ (Disj 𝑥 ∈ {𝐴, 𝐵}𝐶 ↔ ∀𝑦 ∈ {𝐴, 𝐵}∀𝑧 ∈ {𝐴, 𝐵} (𝑦 = 𝑧 ∨ (⦋𝑦 / 𝑥⦌𝐶 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅))
63		pm4.24 565	. 2 ⊢ ((𝐷 ∩ 𝐸) = ∅ ↔ ((𝐷 ∩ 𝐸) = ∅ ∧ (𝐷 ∩ 𝐸) = ∅))
64	61, 62, 63	3bitr4g 314	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ≠ 𝐵) → (Disj 𝑥 ∈ {𝐴, 𝐵}𝐶 ↔ (𝐷 ∩ 𝐸) = ∅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   ∧ w3a 1088   = wceq 1542  ⊤wtru 1543   ∈ wcel 2107   ≠ wne 2941  ∀wral 3062  ⦋csb 3894   ∩ cin 3948  ∅c0 4323  {cpr 4631  Disj wdisj 5114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-nul 4324  df-sn 4630  df-pr 4632  df-disj 5115

This theorem is referenced by:  disjdifprg  31837  pmeasmono  33354  probun  33449  meadjun  45226