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Theorem brcnvep 34664
 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 5496 . . 3 Rel E
21relbrcnv 5760 . 2 (𝐴 E 𝐵𝐵 E 𝐴)
3 epelg 5267 . 2 (𝐴𝑉 → (𝐵 E 𝐴𝐵𝐴))
42, 3syl5bb 275 1 (𝐴𝑉 → (𝐴 E 𝐵𝐵𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∈ wcel 2107   class class class wbr 4886   E cep 5265  ◡ccnv 5354 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4887  df-opab 4949  df-eprel 5266  df-xp 5361  df-rel 5362  df-cnv 5363 This theorem is referenced by:  brcnvepres  34666  eccnvepres  34677  eleccnvep  34678  cnvepres  34697  rnxrncnvepres  34786  dfcoels  34813  br1cossincnvepres  34828  br1cossxrncnvepres  34830
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