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Theorem epelc 4752
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 4751 . 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 setvar class
Syntax hints:    <-> wb 189    e. wcel 1904   _Vcvv 3031   class class class wbr 4395    _E cep 4748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-eprel 4750
This theorem is referenced by:  epel  4753  epini  5204  smoiso  7099  smoiso2  7106  ecid  7446  ordiso2  8048  oismo  8073  cantnflt  8195  cantnfp1lem3  8203  oemapso  8205  cantnflem1b  8209  cantnflem1  8212  cantnf  8216  wemapwe  8220  cnfcomlem  8222  cnfcom  8223  cnfcom3lem  8226  leweon  8460  r0weon  8461  alephiso  8547  fin23lem27  8776  fpwwe2lem9  9081  ex-eprel  25962  dftr6  30461  coep  30462  coepr  30463  brsset  30727  brtxpsd  30732  brcart  30770  dfrecs2  30788  dfrdg4  30789  cnambfre  32053  wepwsolem  35971  dnwech  35977
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