Theorem epelc 4793

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 4792	. 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 184   e. wcel 1767   _Vcvv 3113   class class class wbr 4447   _E cep 4789

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-eprel 4791

This theorem is referenced by:  epel  4794  epini  5367  smoiso  7033  smoiso2  7040  ecid  7376  ordiso2  7940  oismo  7965  cantnflt  8091  cantnfp1lem3  8099  oemapso  8101  cantnflem1b  8105  cantnflem1  8108  cantnf  8112  cantnfltOLD  8121  cantnfp1lem3OLD  8125  cantnflem1bOLD  8128  cantnflem1OLD  8131  cantnfOLD  8134  wemapwe  8139  wemapweOLD  8140  cnfcomlem  8143  cnfcom  8144  cnfcom3lem  8147  cnfcomlemOLD  8151  cnfcomOLD  8152  cnfcom3lemOLD  8155  leweon  8389  r0weon  8390  alephiso  8479  fin23lem27  8708  fpwwe2lem9  9016  ex-eprel  24859  dftr6  28784  coep  28785  coepr  28786  brsset  29144  brtxpsd  29149  brcart  29187  dfrdg4  29205  cnambfre  29668  wepwsolem  30619  dnwech  30626