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Theorem qsdisj 7821
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 2621 . . 3 (𝐴 / 𝑅) = (𝐴 / 𝑅)
3 eqeq1 2625 . . . 4 ([𝑥]𝑅 = 𝐵 → ([𝑥]𝑅 = 𝐶𝐵 = 𝐶))
4 ineq1 3805 . . . . 5 ([𝑥]𝑅 = 𝐵 → ([𝑥]𝑅𝐶) = (𝐵𝐶))
54eqeq1d 2623 . . . 4 ([𝑥]𝑅 = 𝐵 → (([𝑥]𝑅𝐶) = ∅ ↔ (𝐵𝐶) = ∅))
63, 5orbi12d 746 . . 3 ([𝑥]𝑅 = 𝐵 → (([𝑥]𝑅 = 𝐶 ∨ ([𝑥]𝑅𝐶) = ∅) ↔ (𝐵 = 𝐶 ∨ (𝐵𝐶) = ∅)))
7 qsdisj.3 . . . 4 (𝜑𝐶 ∈ (𝐴 / 𝑅))
8 eqeq2 2632 . . . . . 6 ([𝑦]𝑅 = 𝐶 → ([𝑥]𝑅 = [𝑦]𝑅 ↔ [𝑥]𝑅 = 𝐶))
9 ineq2 3806 . . . . . . 7 ([𝑦]𝑅 = 𝐶 → ([𝑥]𝑅 ∩ [𝑦]𝑅) = ([𝑥]𝑅𝐶))
109eqeq1d 2623 . . . . . 6 ([𝑦]𝑅 = 𝐶 → (([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅ ↔ ([𝑥]𝑅𝐶) = ∅))
118, 10orbi12d 746 . . . . 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 762 . . . . . 6 (((𝜑𝑥𝐴) ∧ 𝑦𝐴) → 𝑅 Er 𝑋)
14 erdisj 7791 . . . . . 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 7811 . . . 4 (((𝜑𝑥𝐴) ∧ 𝐶 ∈ (𝐴 / 𝑅)) → ([𝑥]𝑅 = 𝐶 ∨ ([𝑥]𝑅𝐶) = ∅))
177, 16mpidan 704 . . 3 ((𝜑𝑥𝐴) → ([𝑥]𝑅 = 𝐶 ∨ ([𝑥]𝑅𝐶) = ∅))
182, 6, 17ectocld 7811 . 2 ((𝜑𝐵 ∈ (𝐴 / 𝑅)) → (𝐵 = 𝐶 ∨ (𝐵𝐶) = ∅))
191, 18mpdan 702 1 (𝜑 → (𝐵 = 𝐶 ∨ (𝐵𝐶) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 383  wa 384   = wceq 1482  wcel 1989  cin 3571  c0 3913   Er wer 7736  [cec 7737   / cqs 7738
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-br 4652  df-opab 4711  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-er 7739  df-ec 7741  df-qs 7745
This theorem is referenced by:  qsdisj2  7822  uniinqs  7824  cldsubg  21908  erprt  33984
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