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Theorem qsdisj2 8820
Description: A quotient set is a disjoint set. (Contributed by Mario Carneiro, 10-Dec-2016.)
Assertion
Ref Expression
qsdisj2 (𝑅 Er 𝑋Disj 𝑥 ∈ (𝐴 / 𝑅)𝑥)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑋   𝑥,𝑅

Proof of Theorem qsdisj2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 simpl 481 . . . 4 ((𝑅 Er 𝑋 ∧ (𝑥 ∈ (𝐴 / 𝑅) ∧ 𝑦 ∈ (𝐴 / 𝑅))) → 𝑅 Er 𝑋)
2 simprl 769 . . . 4 ((𝑅 Er 𝑋 ∧ (𝑥 ∈ (𝐴 / 𝑅) ∧ 𝑦 ∈ (𝐴 / 𝑅))) → 𝑥 ∈ (𝐴 / 𝑅))
3 simprr 771 . . . 4 ((𝑅 Er 𝑋 ∧ (𝑥 ∈ (𝐴 / 𝑅) ∧ 𝑦 ∈ (𝐴 / 𝑅))) → 𝑦 ∈ (𝐴 / 𝑅))
41, 2, 3qsdisj 8819 . . 3 ((𝑅 Er 𝑋 ∧ (𝑥 ∈ (𝐴 / 𝑅) ∧ 𝑦 ∈ (𝐴 / 𝑅))) → (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
54ralrimivva 3198 . 2 (𝑅 Er 𝑋 → ∀𝑥 ∈ (𝐴 / 𝑅)∀𝑦 ∈ (𝐴 / 𝑅)(𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
6 id 22 . . 3 (𝑥 = 𝑦𝑥 = 𝑦)
76disjor 5132 . 2 (Disj 𝑥 ∈ (𝐴 / 𝑅)𝑥 ↔ ∀𝑥 ∈ (𝐴 / 𝑅)∀𝑦 ∈ (𝐴 / 𝑅)(𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
85, 7sylibr 233 1 (𝑅 Er 𝑋Disj 𝑥 ∈ (𝐴 / 𝑅)𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wo 845   = wceq 1533  wcel 2098  wral 3058  cin 3948  c0 4326  Disj wdisj 5117   Er wer 8728   / cqs 8730
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-disj 5118  df-br 5153  df-opab 5215  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-er 8731  df-ec 8733  df-qs 8737
This theorem is referenced by:  qshash  15813
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