Theorem epel 4432

Description: The epsilon relation and the membership relation are the same. (Contributed by NM, 13-Aug-1995.)

Assertion

Ref	Expression
epel	|- ( x _E y <-> x e. y )

Proof of Theorem epel

Step	Hyp	Ref	Expression
1		vex 2824	. 2 |- y e. _V
2	1	epelc 4431	1 |- ( x _E y <-> x e. y )

Syntax hints:   <-> wb 105   class class class wbr 4125   _E cep 4427

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-eprel 4429

This theorem is referenced by:  epse  4482  wetrep  4500  ordsoexmid  4704  zfregfr  4716  ordwe  4718  wessep  4720  reg3exmidlemwe  4721  smoiso  6563  nnwetri  7213  ordiso2  7365  frec2uzisod  10822  nnti  16936