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Theorem epelc 4410
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 4409 . 2  |-  ( B  e.  _V  ->  ( A  _E  B  <->  A  e.  B ) )
31, 2ax-mp 8 1  |-  ( A  _E  B  <->  A  e.  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    e. wcel 1715   _Vcvv 2873   class class class wbr 4125    _E cep 4406
This theorem is referenced by:  epel  4411  epini  5146  smoiso  6521  smoiso2  6528  ecid  6866  ordiso2  7377  oismo  7402  cantnflt  7520  cantnfp1lem3  7529  oemapso  7531  cantnflem1b  7535  cantnflem1  7538  cantnf  7542  wemapwe  7547  cnfcomlem  7549  cnfcom  7550  cnfcom3lem  7553  leweon  7786  r0weon  7787  alephiso  7872  fin23lem27  8101  fpwwe2lem9  8407  ex-eprel  21252  dftr6  24933  coep  24934  coepr  24935  brsset  25255  brtxpsd  25260  brcart  25297  brcup  25304  brcap  25305  dfrdg4  25315  wepwsolem  26729  dnwech  26736
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-br 4126  df-opab 4180  df-eprel 4408
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