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Theorem epelc 4499
Description: The epsilon relationship and the membership relation are the same. (Contributed by Scott Fenton, 11-Apr-2012.)
Hypothesis
Ref Expression
epelc.1  |-  B  e. 
_V
Assertion
Ref Expression
epelc  |-  ( A  _E  B  <->  A  e.  B )

Proof of Theorem epelc
StepHypRef Expression
1 epelc.1 . 2  |-  B  e. 
_V
2 epelg 4498 . 2  |-  ( B  e.  _V  ->  ( A  _E  B  <->  A  e.  B ) )
31, 2ax-mp 5 1  |-  ( A  _E  B  <->  A  e.  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    e. wcel 1726   _Vcvv 2958   class class class wbr 4215    _E cep 4495
This theorem is referenced by:  epel  4500  epini  5237  smoiso  6627  smoiso2  6634  ecid  6972  ordiso2  7487  oismo  7512  cantnflt  7630  cantnfp1lem3  7639  oemapso  7641  cantnflem1b  7645  cantnflem1  7648  cantnf  7652  wemapwe  7657  cnfcomlem  7659  cnfcom  7660  cnfcom3lem  7663  leweon  7898  r0weon  7899  alephiso  7984  fin23lem27  8213  fpwwe2lem9  8518  ex-eprel  21746  dftr6  25378  coep  25379  coepr  25380  brsset  25739  brtxpsd  25744  brcart  25782  dfrdg4  25800  cnambfre  26267  wepwsolem  27130  dnwech  27137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-eprel 4497
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