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Theorem brcnvep 38247
Description: The converse of the binary epsilon relation. (Contributed by Peter Mazsa, 30-Jan-2018.)
Assertion
Ref Expression
brcnvep (𝐴𝑉 → (𝐴 E 𝐵𝐵𝐴))

Proof of Theorem brcnvep
StepHypRef Expression
1 rele 5840 . . 3 Rel E
21relbrcnv 6128 . 2 (𝐴 E 𝐵𝐵 E 𝐴)
3 epelg 5590 . 2 (𝐴𝑉 → (𝐵 E 𝐴𝐵𝐴))
42, 3bitrid 283 1 (𝐴𝑉 → (𝐴 E 𝐵𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wcel 2106   class class class wbr 5148   E cep 5588  ccnv 5688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-eprel 5589  df-xp 5695  df-rel 5696  df-cnv 5697
This theorem is referenced by:  brcnvepres  38249  eccnvepres  38262  eleccnvep  38263  cnvepres  38280  rnxrncnvepres  38382  coss2cnvepres  38400  dfcoels  38412  br1cossincnvepres  38432  br1cossxrncnvepres  38434  dfeldisj5  38703
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