Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  epel GIF version

Theorem epel 4182
 Description: The epsilon relation and the membership relation are the same. (Contributed by NM, 13-Aug-1995.)
Assertion
Ref Expression
epel (𝑥 E 𝑦𝑥𝑦)

Proof of Theorem epel
StepHypRef Expression
1 vex 2661 . 2 𝑦 ∈ V
21epelc 4181 1 (𝑥 E 𝑦𝑥𝑦)
 Colors of variables: wff set class Syntax hints:   ↔ wb 104   class class class wbr 3897   E cep 4177 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099 This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-eprel 4179 This theorem is referenced by:  epse  4232  wetrep  4250  ordsoexmid  4445  zfregfr  4456  ordwe  4458  wessep  4460  reg3exmidlemwe  4461  smoiso  6165  nnwetri  6770  ordiso2  6886  frec2uzisod  10120  nnti  12993
 Copyright terms: Public domain W3C validator