Theorem qsdisj 8816

Description: Members of a quotient set do not overlap. (Contributed by Rodolfo Medina, 12-Oct-2010.) (Revised by Mario Carneiro, 11-Jul-2014.)

Hypotheses

Ref	Expression
qsdisj.1	⊢ (𝜑 → 𝑅 Er 𝑋)
qsdisj.2	⊢ (𝜑 → 𝐵 ∈ (𝐴 / 𝑅))
qsdisj.3	⊢ (𝜑 → 𝐶 ∈ (𝐴 / 𝑅))

Assertion

Ref	Expression
qsdisj	⊢ (𝜑 → (𝐵 = 𝐶 ∨ (𝐵 ∩ 𝐶) = ∅))

Proof of Theorem qsdisj

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

Step	Hyp	Ref	Expression
1		qsdisj.2	. 2 ⊢ (𝜑 → 𝐵 ∈ (𝐴 / 𝑅))
2		eqid 2734	. . 3 ⊢ (𝐴 / 𝑅) = (𝐴 / 𝑅)
3		eqeq1 2738	. . . 4 ⊢ ([𝑥]𝑅 = 𝐵 → ([𝑥]𝑅 = 𝐶 ↔ 𝐵 = 𝐶))
4		ineq1 4193	. . . . 5 ⊢ ([𝑥]𝑅 = 𝐵 → ([𝑥]𝑅 ∩ 𝐶) = (𝐵 ∩ 𝐶))
5	4	eqeq1d 2736	. . . 4 ⊢ ([𝑥]𝑅 = 𝐵 → (([𝑥]𝑅 ∩ 𝐶) = ∅ ↔ (𝐵 ∩ 𝐶) = ∅))
6	3, 5	orbi12d 918	. . 3 ⊢ ([𝑥]𝑅 = 𝐵 → (([𝑥]𝑅 = 𝐶 ∨ ([𝑥]𝑅 ∩ 𝐶) = ∅) ↔ (𝐵 = 𝐶 ∨ (𝐵 ∩ 𝐶) = ∅)))
7		qsdisj.3	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐴 / 𝑅))
8		eqeq2 2746	. . . . . 6 ⊢ ([𝑦]𝑅 = 𝐶 → ([𝑥]𝑅 = [𝑦]𝑅 ↔ [𝑥]𝑅 = 𝐶))
9		ineq2 4194	. . . . . . 7 ⊢ ([𝑦]𝑅 = 𝐶 → ([𝑥]𝑅 ∩ [𝑦]𝑅) = ([𝑥]𝑅 ∩ 𝐶))
10	9	eqeq1d 2736	. . . . . 6 ⊢ ([𝑦]𝑅 = 𝐶 → (([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅ ↔ ([𝑥]𝑅 ∩ 𝐶) = ∅))
11	8, 10	orbi12d 918	. . . . 5 ⊢ ([𝑦]𝑅 = 𝐶 → (([𝑥]𝑅 = [𝑦]𝑅 ∨ ([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅) ↔ ([𝑥]𝑅 = 𝐶 ∨ ([𝑥]𝑅 ∩ 𝐶) = ∅)))
12		qsdisj.1	. . . . . . 7 ⊢ (𝜑 → 𝑅 Er 𝑋)
13	12	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑅 Er 𝑋)
14		erdisj 8781	. . . . . 6 ⊢ (𝑅 Er 𝑋 → ([𝑥]𝑅 = [𝑦]𝑅 ∨ ([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅))
15	13, 14	syl 17	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → ([𝑥]𝑅 = [𝑦]𝑅 ∨ ([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅))
16	2, 11, 15	ectocld 8806	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝐶 ∈ (𝐴 / 𝑅)) → ([𝑥]𝑅 = 𝐶 ∨ ([𝑥]𝑅 ∩ 𝐶) = ∅))
17	7, 16	mpidan 689	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ([𝑥]𝑅 = 𝐶 ∨ ([𝑥]𝑅 ∩ 𝐶) = ∅))
18	2, 6, 17	ectocld 8806	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐴 / 𝑅)) → (𝐵 = 𝐶 ∨ (𝐵 ∩ 𝐶) = ∅))
19	1, 18	mpdan 687	1 ⊢ (𝜑 → (𝐵 = 𝐶 ∨ (𝐵 ∩ 𝐶) = ∅))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1539   ∈ wcel 2107   ∩ cin 3930  ∅c0 4313   Er wer 8724  [cec 8725   / cqs 8726

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  ax-sep 5276  ax-nul 5286  ax-pr 5412

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-clab 2713  df-cleq 2726  df-clel 2808  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5124  df-opab 5186  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-er 8727  df-ec 8729  df-qs 8733

This theorem is referenced by:  qsdisj2  8817  uniinqs  8819  cldsubg  24065  erprt  38833