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Theorem qsdisjALTV 38597
Description: Elements of a quotient set do not overlap. (Contributed by Rodolfo Medina, 12-Oct-2010.) (Revised by Mario Carneiro, 11-Jul-2014.) (Revised by Peter Mazsa, 3-Jun-2019.)
Hypotheses
Ref Expression
qsdisjALTV.1 (𝜑 → EqvRel 𝑅)
qsdisjALTV.2 (𝜑𝐵 ∈ (𝐴 / 𝑅))
qsdisjALTV.3 (𝜑𝐶 ∈ (𝐴 / 𝑅))
Assertion
Ref Expression
qsdisjALTV (𝜑 → (𝐵 = 𝐶 ∨ (𝐵𝐶) = ∅))

Proof of Theorem qsdisjALTV
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 qsdisjALTV.2 . 2 (𝜑𝐵 ∈ (𝐴 / 𝑅))
2 eqid 2735 . . 3 (𝐴 / 𝑅) = (𝐴 / 𝑅)
3 eqeq1 2739 . . . 4 ([𝑥]𝑅 = 𝐵 → ([𝑥]𝑅 = 𝐶𝐵 = 𝐶))
4 ineq1 4221 . . . . 5 ([𝑥]𝑅 = 𝐵 → ([𝑥]𝑅𝐶) = (𝐵𝐶))
54eqeq1d 2737 . . . 4 ([𝑥]𝑅 = 𝐵 → (([𝑥]𝑅𝐶) = ∅ ↔ (𝐵𝐶) = ∅))
63, 5orbi12d 918 . . 3 ([𝑥]𝑅 = 𝐵 → (([𝑥]𝑅 = 𝐶 ∨ ([𝑥]𝑅𝐶) = ∅) ↔ (𝐵 = 𝐶 ∨ (𝐵𝐶) = ∅)))
7 qsdisjALTV.3 . . . 4 (𝜑𝐶 ∈ (𝐴 / 𝑅))
8 eqeq2 2747 . . . . . 6 ([𝑦]𝑅 = 𝐶 → ([𝑥]𝑅 = [𝑦]𝑅 ↔ [𝑥]𝑅 = 𝐶))
9 ineq2 4222 . . . . . . 7 ([𝑦]𝑅 = 𝐶 → ([𝑥]𝑅 ∩ [𝑦]𝑅) = ([𝑥]𝑅𝐶))
109eqeq1d 2737 . . . . . 6 ([𝑦]𝑅 = 𝐶 → (([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅ ↔ ([𝑥]𝑅𝐶) = ∅))
118, 10orbi12d 918 . . . . 5 ([𝑦]𝑅 = 𝐶 → (([𝑥]𝑅 = [𝑦]𝑅 ∨ ([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅) ↔ ([𝑥]𝑅 = 𝐶 ∨ ([𝑥]𝑅𝐶) = ∅)))
12 qsdisjALTV.1 . . . . . . 7 (𝜑 → EqvRel 𝑅)
1312ad2antrr 726 . . . . . 6 (((𝜑𝑥𝐴) ∧ 𝑦𝐴) → EqvRel 𝑅)
14 eqvreldisj 38596 . . . . . 6 ( EqvRel 𝑅 → ([𝑥]𝑅 = [𝑦]𝑅 ∨ ([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅))
1513, 14syl 17 . . . . 5 (((𝜑𝑥𝐴) ∧ 𝑦𝐴) → ([𝑥]𝑅 = [𝑦]𝑅 ∨ ([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅))
162, 11, 15ectocld 8823 . . . 4 (((𝜑𝑥𝐴) ∧ 𝐶 ∈ (𝐴 / 𝑅)) → ([𝑥]𝑅 = 𝐶 ∨ ([𝑥]𝑅𝐶) = ∅))
177, 16mpidan 689 . . 3 ((𝜑𝑥𝐴) → ([𝑥]𝑅 = 𝐶 ∨ ([𝑥]𝑅𝐶) = ∅))
182, 6, 17ectocld 8823 . 2 ((𝜑𝐵 ∈ (𝐴 / 𝑅)) → (𝐵 = 𝐶 ∨ (𝐵𝐶) = ∅))
191, 18mpdan 687 1 (𝜑 → (𝐵 = 𝐶 ∨ (𝐵𝐶) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 847   = wceq 1537  wcel 2106  cin 3962  c0 4339  [cec 8742   / cqs 8743   EqvRel weqvrel 38179
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ec 8746  df-qs 8750  df-refrel 38494  df-symrel 38526  df-trrel 38556  df-eqvrel 38567
This theorem is referenced by:  eqvreldisj1  38806
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