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Theorem qsdisj 8363
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 2818 . . 3 (𝐴 / 𝑅) = (𝐴 / 𝑅)
3 eqeq1 2822 . . . 4 ([𝑥]𝑅 = 𝐵 → ([𝑥]𝑅 = 𝐶𝐵 = 𝐶))
4 ineq1 4178 . . . . 5 ([𝑥]𝑅 = 𝐵 → ([𝑥]𝑅𝐶) = (𝐵𝐶))
54eqeq1d 2820 . . . 4 ([𝑥]𝑅 = 𝐵 → (([𝑥]𝑅𝐶) = ∅ ↔ (𝐵𝐶) = ∅))
63, 5orbi12d 912 . . 3 ([𝑥]𝑅 = 𝐵 → (([𝑥]𝑅 = 𝐶 ∨ ([𝑥]𝑅𝐶) = ∅) ↔ (𝐵 = 𝐶 ∨ (𝐵𝐶) = ∅)))
7 qsdisj.3 . . . 4 (𝜑𝐶 ∈ (𝐴 / 𝑅))
8 eqeq2 2830 . . . . . 6 ([𝑦]𝑅 = 𝐶 → ([𝑥]𝑅 = [𝑦]𝑅 ↔ [𝑥]𝑅 = 𝐶))
9 ineq2 4180 . . . . . . 7 ([𝑦]𝑅 = 𝐶 → ([𝑥]𝑅 ∩ [𝑦]𝑅) = ([𝑥]𝑅𝐶))
109eqeq1d 2820 . . . . . 6 ([𝑦]𝑅 = 𝐶 → (([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅ ↔ ([𝑥]𝑅𝐶) = ∅))
118, 10orbi12d 912 . . . . 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 722 . . . . . 6 (((𝜑𝑥𝐴) ∧ 𝑦𝐴) → 𝑅 Er 𝑋)
14 erdisj 8330 . . . . . 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 8353 . . . 4 (((𝜑𝑥𝐴) ∧ 𝐶 ∈ (𝐴 / 𝑅)) → ([𝑥]𝑅 = 𝐶 ∨ ([𝑥]𝑅𝐶) = ∅))
177, 16mpidan 685 . . 3 ((𝜑𝑥𝐴) → ([𝑥]𝑅 = 𝐶 ∨ ([𝑥]𝑅𝐶) = ∅))
182, 6, 17ectocld 8353 . 2 ((𝜑𝐵 ∈ (𝐴 / 𝑅)) → (𝐵 = 𝐶 ∨ (𝐵𝐶) = ∅))
191, 18mpdan 683 1 (𝜑 → (𝐵 = 𝐶 ∨ (𝐵𝐶) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wo 841   = wceq 1528  wcel 2105  cin 3932  c0 4288   Er wer 8275  [cec 8276   / cqs 8277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-er 8278  df-ec 8280  df-qs 8284
This theorem is referenced by:  qsdisj2  8364  uniinqs  8366  cldsubg  22646  erprt  35889
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