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Theorem brcnvep 37769
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 5833 . . 3 Rel E
21relbrcnv 6116 . 2 (𝐴 E 𝐵𝐵 E 𝐴)
3 epelg 5587 . 2 (𝐴𝑉 → (𝐵 E 𝐴𝐵𝐴))
42, 3bitrid 282 1 (𝐴𝑉 → (𝐴 E 𝐵𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2098   class class class wbr 5152   E cep 5585  ccnv 5681
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 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-eprel 5586  df-xp 5688  df-rel 5689  df-cnv 5690
This theorem is referenced by:  brcnvepres  37771  eccnvepres  37784  eleccnvep  37785  cnvepres  37802  rnxrncnvepres  37904  coss2cnvepres  37922  dfcoels  37934  br1cossincnvepres  37954  br1cossxrncnvepres  37956  dfeldisj5  38225
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