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Theorem brcnvep 37644
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 5820 . . 3 Rel E
21relbrcnv 6099 . 2 (𝐴 E 𝐵𝐵 E 𝐴)
3 epelg 5574 . 2 (𝐴𝑉 → (𝐵 E 𝐴𝐵𝐴))
42, 3bitrid 283 1 (𝐴𝑉 → (𝐴 E 𝐵𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2098   class class class wbr 5141   E cep 5572  ccnv 5668
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-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-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  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-eprel 5573  df-xp 5675  df-rel 5676  df-cnv 5677
This theorem is referenced by:  brcnvepres  37646  eccnvepres  37659  eleccnvep  37660  cnvepres  37678  rnxrncnvepres  37781  coss2cnvepres  37799  dfcoels  37811  br1cossincnvepres  37831  br1cossxrncnvepres  37833  dfeldisj5  38102
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