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Theorem brcnvep 37741
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 5831 . . 3 Rel E
21relbrcnv 6114 . 2 (𝐴 E 𝐵𝐵 E 𝐴)
3 epelg 5585 . 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 5150   E cep 5583  ccnv 5679
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 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431
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 2705  df-cleq 2719  df-clel 2805  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5151  df-opab 5213  df-eprel 5584  df-xp 5686  df-rel 5687  df-cnv 5688
This theorem is referenced by:  brcnvepres  37743  eccnvepres  37756  eleccnvep  37757  cnvepres  37774  rnxrncnvepres  37876  coss2cnvepres  37894  dfcoels  37906  br1cossincnvepres  37926  br1cossxrncnvepres  37928  dfeldisj5  38197
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