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Theorem eqvreldisj2 38208
Description: The elements of the quotient set of an equivalence relation are disjoint (cf. eqvreldisj3 38209). (Contributed by Mario Carneiro, 10-Dec-2016.) (Revised by Peter Mazsa, 19-Sep-2021.)
Assertion
Ref Expression
eqvreldisj2 ( EqvRel 𝑅 → ElDisj (𝐴 / 𝑅))

Proof of Theorem eqvreldisj2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqvreldisj1 38207 . 2 ( EqvRel 𝑅 → ∀𝑥 ∈ (𝐴 / 𝑅)∀𝑦 ∈ (𝐴 / 𝑅)(𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
2 dfeldisj5 38104 . 2 ( ElDisj (𝐴 / 𝑅) ↔ ∀𝑥 ∈ (𝐴 / 𝑅)∀𝑦 ∈ (𝐴 / 𝑅)(𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
31, 2sylibr 233 1 ( EqvRel 𝑅 → ElDisj (𝐴 / 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 844   = wceq 1533  wral 3055  cin 3942  c0 4317   / cqs 8704   EqvRel weqvrel 37573   ElDisj weldisj 37592
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-id 5567  df-eprel 5573  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ec 8707  df-qs 8711  df-coss 37794  df-refrel 37895  df-cnvrefrel 37910  df-symrel 37927  df-trrel 37957  df-eqvrel 37968  df-disjALTV 38088  df-eldisj 38090
This theorem is referenced by:  eqvreldisj3  38209  eqvrelqseqdisj2  38212
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