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Theorem epelc 4307
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 4306 . 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 1684   _Vcvv 2788   class class class wbr 4023    _E cep 4303
This theorem is referenced by:  epel  4308  epini  5043  smoiso  6379  smoiso2  6386  ecid  6724  ordiso2  7230  oismo  7255  cantnflt  7373  cantnfp1lem3  7382  oemapso  7384  cantnflem1b  7388  cantnflem1  7391  cantnf  7395  wemapwe  7400  cnfcomlem  7402  cnfcom  7403  cnfcom3lem  7406  leweon  7639  r0weon  7640  alephiso  7725  fin23lem27  7954  fpwwe2lem9  8260  ex-eprel  20820  dftr6  24107  coep  24108  coepr  24109  brsset  24429  brtxpsd  24434  brcart  24471  brcup  24478  brcap  24479  dfrdg4  24488  wepwsolem  27138  dnwech  27145
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-eprel 4305
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