MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  qsdisj Structured version   Visualization version   GIF version

Theorem qsdisj 8369
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 2826 . . 3 (𝐴 / 𝑅) = (𝐴 / 𝑅)
3 eqeq1 2830 . . . 4 ([𝑥]𝑅 = 𝐵 → ([𝑥]𝑅 = 𝐶𝐵 = 𝐶))
4 ineq1 4185 . . . . 5 ([𝑥]𝑅 = 𝐵 → ([𝑥]𝑅𝐶) = (𝐵𝐶))
54eqeq1d 2828 . . . 4 ([𝑥]𝑅 = 𝐵 → (([𝑥]𝑅𝐶) = ∅ ↔ (𝐵𝐶) = ∅))
63, 5orbi12d 914 . . 3 ([𝑥]𝑅 = 𝐵 → (([𝑥]𝑅 = 𝐶 ∨ ([𝑥]𝑅𝐶) = ∅) ↔ (𝐵 = 𝐶 ∨ (𝐵𝐶) = ∅)))
7 qsdisj.3 . . . 4 (𝜑𝐶 ∈ (𝐴 / 𝑅))
8 eqeq2 2838 . . . . . 6 ([𝑦]𝑅 = 𝐶 → ([𝑥]𝑅 = [𝑦]𝑅 ↔ [𝑥]𝑅 = 𝐶))
9 ineq2 4187 . . . . . . 7 ([𝑦]𝑅 = 𝐶 → ([𝑥]𝑅 ∩ [𝑦]𝑅) = ([𝑥]𝑅𝐶))
109eqeq1d 2828 . . . . . 6 ([𝑦]𝑅 = 𝐶 → (([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅ ↔ ([𝑥]𝑅𝐶) = ∅))
118, 10orbi12d 914 . . . . 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 8336 . . . . . 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 8359 . . . 4 (((𝜑𝑥𝐴) ∧ 𝐶 ∈ (𝐴 / 𝑅)) → ([𝑥]𝑅 = 𝐶 ∨ ([𝑥]𝑅𝐶) = ∅))
177, 16mpidan 685 . . 3 ((𝜑𝑥𝐴) → ([𝑥]𝑅 = 𝐶 ∨ ([𝑥]𝑅𝐶) = ∅))
182, 6, 17ectocld 8359 . 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 843   = wceq 1530  wcel 2107  cin 3939  c0 4295   Er wer 8281  [cec 8282   / cqs 8283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-br 5064  df-opab 5126  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-er 8284  df-ec 8286  df-qs 8290
This theorem is referenced by:  qsdisj2  8370  uniinqs  8372  cldsubg  22653  erprt  35895
  Copyright terms: Public domain W3C validator