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Theorem eqvreldisj4 38770
Description: Intersection with the converse epsilon relation restricted to the quotient set of an equivalence relation is disjoint. (Contributed by Peter Mazsa, 30-May-2020.) (Revised by Peter Mazsa, 31-Dec-2021.)
Assertion
Ref Expression
eqvreldisj4 ( EqvRel 𝑅 → Disj (𝑆 ∩ ( E ↾ (𝐵 / 𝑅))))

Proof of Theorem eqvreldisj4
StepHypRef Expression
1 eqvreldisj3 38769 . 2 ( EqvRel 𝑅 → Disj ( E ↾ (𝐵 / 𝑅)))
2 disjimin 38694 . 2 ( Disj ( E ↾ (𝐵 / 𝑅)) → Disj (𝑆 ∩ ( E ↾ (𝐵 / 𝑅))))
31, 2syl 17 1 ( EqvRel 𝑅 → Disj (𝑆 ∩ ( E ↾ (𝐵 / 𝑅))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  cin 3962   E cep 5581  ccnv 5682  cres 5685   / cqs 8737   EqvRel weqvrel 38139   Disj wdisjALTV 38156
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-10 2137  ax-11 2153  ax-12 2173  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5430
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2536  df-eu 2565  df-clab 2711  df-cleq 2725  df-clel 2812  df-nfc 2888  df-ne 2937  df-ral 3058  df-rex 3067  df-rmo 3376  df-rab 3433  df-v 3479  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150  df-opab 5212  df-id 5576  df-eprel 5582  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-ec 8740  df-qs 8744  df-coss 38354  df-refrel 38455  df-cnvrefrel 38470  df-symrel 38487  df-trrel 38517  df-eqvrel 38528  df-funALTV 38625  df-disjALTV 38648  df-eldisj 38650
This theorem is referenced by: (None)
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