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Theorem qsdisj 8813
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 (𝜑𝑅 Er 𝑋)
qsdisj.2 (𝜑𝐵 ∈ (𝐴 / 𝑅))
qsdisj.3 (𝜑𝐶 ∈ (𝐴 / 𝑅))
Assertion
Ref Expression
qsdisj (𝜑 → (𝐵 = 𝐶 ∨ (𝐵𝐶) = ∅))

Proof of Theorem qsdisj
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 qsdisj.2 . 2 (𝜑𝐵 ∈ (𝐴 / 𝑅))
2 eqid 2728 . . 3 (𝐴 / 𝑅) = (𝐴 / 𝑅)
3 eqeq1 2732 . . . 4 ([𝑥]𝑅 = 𝐵 → ([𝑥]𝑅 = 𝐶𝐵 = 𝐶))
4 ineq1 4205 . . . . 5 ([𝑥]𝑅 = 𝐵 → ([𝑥]𝑅𝐶) = (𝐵𝐶))
54eqeq1d 2730 . . . 4 ([𝑥]𝑅 = 𝐵 → (([𝑥]𝑅𝐶) = ∅ ↔ (𝐵𝐶) = ∅))
63, 5orbi12d 917 . . 3 ([𝑥]𝑅 = 𝐵 → (([𝑥]𝑅 = 𝐶 ∨ ([𝑥]𝑅𝐶) = ∅) ↔ (𝐵 = 𝐶 ∨ (𝐵𝐶) = ∅)))
7 qsdisj.3 . . . 4 (𝜑𝐶 ∈ (𝐴 / 𝑅))
8 eqeq2 2740 . . . . . 6 ([𝑦]𝑅 = 𝐶 → ([𝑥]𝑅 = [𝑦]𝑅 ↔ [𝑥]𝑅 = 𝐶))
9 ineq2 4206 . . . . . . 7 ([𝑦]𝑅 = 𝐶 → ([𝑥]𝑅 ∩ [𝑦]𝑅) = ([𝑥]𝑅𝐶))
109eqeq1d 2730 . . . . . 6 ([𝑦]𝑅 = 𝐶 → (([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅ ↔ ([𝑥]𝑅𝐶) = ∅))
118, 10orbi12d 917 . . . . 5 ([𝑦]𝑅 = 𝐶 → (([𝑥]𝑅 = [𝑦]𝑅 ∨ ([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅) ↔ ([𝑥]𝑅 = 𝐶 ∨ ([𝑥]𝑅𝐶) = ∅)))
12 qsdisj.1 . . . . . . 7 (𝜑𝑅 Er 𝑋)
1312ad2antrr 725 . . . . . 6 (((𝜑𝑥𝐴) ∧ 𝑦𝐴) → 𝑅 Er 𝑋)
14 erdisj 8778 . . . . . 6 (𝑅 Er 𝑋 → ([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 8803 . . . 4 (((𝜑𝑥𝐴) ∧ 𝐶 ∈ (𝐴 / 𝑅)) → ([𝑥]𝑅 = 𝐶 ∨ ([𝑥]𝑅𝐶) = ∅))
177, 16mpidan 688 . . 3 ((𝜑𝑥𝐴) → ([𝑥]𝑅 = 𝐶 ∨ ([𝑥]𝑅𝐶) = ∅))
182, 6, 17ectocld 8803 . 2 ((𝜑𝐵 ∈ (𝐴 / 𝑅)) → (𝐵 = 𝐶 ∨ (𝐵𝐶) = ∅))
191, 18mpdan 686 1 (𝜑 → (𝐵 = 𝐶 ∨ (𝐵𝐶) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 846   = wceq 1534  wcel 2099  cin 3946  c0 4323   Er wer 8722  [cec 8723   / cqs 8724
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-er 8725  df-ec 8727  df-qs 8731
This theorem is referenced by:  qsdisj2  8814  uniinqs  8816  cldsubg  24028  erprt  38345
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