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Theorem brcnvep 36404
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 5737 . . 3 Rel E
21relbrcnv 6015 . 2 (𝐴 E 𝐵𝐵 E 𝐴)
3 epelg 5496 . 2 (𝐴𝑉 → (𝐵 E 𝐴𝐵𝐴))
42, 3syl5bb 283 1 (𝐴𝑉 → (𝐴 E 𝐵𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2106   class class class wbr 5074   E cep 5494  ccnv 5588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-eprel 5495  df-xp 5595  df-rel 5596  df-cnv 5597
This theorem is referenced by:  brcnvepres  36406  eccnvepres  36415  eleccnvep  36416  cnvepres  36433  rnxrncnvepres  36526  dfcoels  36553  br1cossincnvepres  36568  br1cossxrncnvepres  36570  dfeldisj5  36832
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