Theorem disjprg 5119

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 2738	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑦 = 𝑧 ↔ 𝐴 = 𝑧))
2		nfcv 2897	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝐴
3		nfcv 2897	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝐷
4		disjprg.1	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → 𝐶 = 𝐷)
5	2, 3, 4	csbhypf 3907	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐶 = 𝐷)
6	5	ineq1d 4199	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (⦋𝑦 / 𝑥⦌𝐶 ∩ ⦋𝑧 / 𝑥⦌𝐶) = (𝐷 ∩ ⦋𝑧 / 𝑥⦌𝐶))
7	6	eqeq1d 2736	. . . . . . 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 3165	. . . . 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 2738	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑦 = 𝑧 ↔ 𝐵 = 𝑧))
11		nfcv 2897	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝐵
12		nfcv 2897	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝐸
13		disjprg.2	. . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → 𝐶 = 𝐸)
14	11, 12, 13	csbhypf 3907	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → ⦋𝑦 / 𝑥⦌𝐶 = 𝐸)
15	14	ineq1d 4199	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (⦋𝑦 / 𝑥⦌𝐶 ∩ ⦋𝑧 / 𝑥⦌𝐶) = (𝐸 ∩ ⦋𝑧 / 𝑥⦌𝐶))
16	15	eqeq1d 2736	. . . . . . 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 3165	. . . . 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 4676	. . . 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 1132	. . 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 2740	. . . . . . . . 9 ⊢ (𝑧 = 𝐴 → 𝐴 = 𝑧)
23	22	orcd 873	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → (𝐴 = 𝑧 ∨ (𝐷 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅))
24		trud 1549	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → ⊤)
25	23, 24	2thd 265	. . . . . . 7 ⊢ (𝑧 = 𝐴 → ((𝐴 = 𝑧 ∨ (𝐷 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅) ↔ ⊤))
26		eqeq2 2746	. . . . . . . 8 ⊢ (𝑧 = 𝐵 → (𝐴 = 𝑧 ↔ 𝐴 = 𝐵))
27	11, 12, 13	csbhypf 3907	. . . . . . . . . 10 ⊢ (𝑧 = 𝐵 → ⦋𝑧 / 𝑥⦌𝐶 = 𝐸)
28	27	ineq2d 4200	. . . . . . . . 9 ⊢ (𝑧 = 𝐵 → (𝐷 ∩ ⦋𝑧 / 𝑥⦌𝐶) = (𝐷 ∩ 𝐸))
29	28	eqeq1d 2736	. . . . . . . 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 4676	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (∀𝑧 ∈ {𝐴, 𝐵} (𝐴 = 𝑧 ∨ (𝐷 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅) ↔ (⊤ ∧ (𝐴 = 𝐵 ∨ (𝐷 ∩ 𝐸) = ∅))))
32	31	3adant3 1132	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ≠ 𝐵) → (∀𝑧 ∈ {𝐴, 𝐵} (𝐴 = 𝑧 ∨ (𝐷 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅) ↔ (⊤ ∧ (𝐴 = 𝐵 ∨ (𝐷 ∩ 𝐸) = ∅))))
33		simp3 1138	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ≠ 𝐵) → 𝐴 ≠ 𝐵)
34	33	neneqd 2936	. . . . . . 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 1543	. . . . . . 7 ⊢ ⊤
38	37	biantrur 530	. . . . . 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 2746	. . . . . . . . 9 ⊢ (𝑧 = 𝐴 → (𝐵 = 𝑧 ↔ 𝐵 = 𝐴))
42		eqcom 2741	. . . . . . . . 9 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵)
43	41, 42	bitrdi 287	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → (𝐵 = 𝑧 ↔ 𝐴 = 𝐵))
44	2, 3, 4	csbhypf 3907	. . . . . . . . . . 11 ⊢ (𝑧 = 𝐴 → ⦋𝑧 / 𝑥⦌𝐶 = 𝐷)
45	44	ineq2d 4200	. . . . . . . . . 10 ⊢ (𝑧 = 𝐴 → (𝐸 ∩ ⦋𝑧 / 𝑥⦌𝐶) = (𝐸 ∩ 𝐷))
46		incom 4189	. . . . . . . . . 10 ⊢ (𝐸 ∩ 𝐷) = (𝐷 ∩ 𝐸)
47	45, 46	eqtrdi 2785	. . . . . . . . 9 ⊢ (𝑧 = 𝐴 → (𝐸 ∩ ⦋𝑧 / 𝑥⦌𝐶) = (𝐷 ∩ 𝐸))
48	47	eqeq1d 2736	. . . . . . . 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 2740	. . . . . . . . 9 ⊢ (𝑧 = 𝐵 → 𝐵 = 𝑧)
52	51	orcd 873	. . . . . . . 8 ⊢ (𝑧 = 𝐵 → (𝐵 = 𝑧 ∨ (𝐸 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅))
53		trud 1549	. . . . . . . 8 ⊢ (𝑧 = 𝐵 → ⊤)
54	52, 53	2thd 265	. . . . . . 7 ⊢ (𝑧 = 𝐵 → ((𝐵 = 𝑧 ∨ (𝐸 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅) ↔ ⊤))
55	49, 54	ralprg 4676	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (∀𝑧 ∈ {𝐴, 𝐵} (𝐵 = 𝑧 ∨ (𝐸 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅) ↔ ((𝐴 = 𝐵 ∨ (𝐷 ∩ 𝐸) = ∅) ∧ ⊤)))
56	55	3adant3 1132	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ≠ 𝐵) → (∀𝑧 ∈ {𝐴, 𝐵} (𝐵 = 𝑧 ∨ (𝐸 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅) ↔ ((𝐴 = 𝐵 ∨ (𝐷 ∩ 𝐸) = ∅) ∧ ⊤)))
57	37	biantru 529	. . . . . 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 5106	. 2 ⊢ (Disj 𝑥 ∈ {𝐴, 𝐵}𝐶 ↔ ∀𝑦 ∈ {𝐴, 𝐵}∀𝑧 ∈ {𝐴, 𝐵} (𝑦 = 𝑧 ∨ (⦋𝑦 / 𝑥⦌𝐶 ∩ ⦋𝑧 / 𝑥⦌𝐶) = ∅))
63		pm4.24 563	. 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 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1539  ⊤wtru 1540   ∈ wcel 2107   ≠ wne 2931  ∀wral 3050  ⦋csb 3879   ∩ cin 3930  ∅c0 4313  {cpr 4608  Disj wdisj 5090

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3363  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-nul 4314  df-sn 4607  df-pr 4609  df-disj 5091

This theorem is referenced by:  disjdifprg  32523  unelldsys  34118  pmeasmono  34285  probun  34380  meadjun  46434