Theorem epelc 5031

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

Step	Hyp	Ref	Expression
1		epelc.1	. 2 |- B e. _V
2		epelg 5030	. 2 |- ( B e. _V -> ( A _E B <-> A e. B ) )
3	1, 2	ax-mp 5	1 |- ( A _E B <-> A e. B )

Syntax hints:   <-> wb 196   e. wcel 1990   _Vcvv 3200   class class class wbr 4653   _E cep 5028

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-eprel 5029

This theorem is referenced by:  epel  5032  epini  5495  smoiso  7459  smoiso2  7466  ecid  7812  ordiso2  8420  oismo  8445  cantnflt  8569  cantnfp1lem3  8577  oemapso  8579  cantnflem1b  8583  cantnflem1  8586  cantnf  8590  wemapwe  8594  cnfcomlem  8596  cnfcom  8597  cnfcom3lem  8600  leweon  8834  r0weon  8835  alephiso  8921  fin23lem27  9150  fpwwe2lem9  9460  ex-eprel  27290  dftr6  31640  coep  31641  coepr  31642  brsset  31996  brtxpsd  32001  brcart  32039  dfrecs2  32057  dfrdg4  32058  cnambfre  33458  wepwsolem  37612  dnwech  37618