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Theorem ideq 4965
Description: For sets, the identity relation is the same as equality. (Contributed by NM, 13-Aug-1995.)
Hypothesis
Ref Expression
ideq.1  |-  B  e. 
_V
Assertion
Ref Expression
ideq  |-  ( A  _I  B  <->  A  =  B )

Proof of Theorem ideq
StepHypRef Expression
1 ideq.1 . 2  |-  B  e. 
_V
2 ideqg 4964 . 2  |-  ( B  e.  _V  ->  ( A  _I  B  <->  A  =  B ) )
31, 2ax-mp 8 1  |-  ( A  _I  B  <->  A  =  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1649    e. wcel 1717   _Vcvv 2899   class class class wbr 4153    _I cid 4434
This theorem is referenced by:  dmi  5024  resieq  5096  resiexg  5128  iss  5129  imai  5158  issref  5187  intasym  5189  asymref  5190  intirr  5192  poirr2  5198  cnvi  5216  coi1  5325  dffv2  5735  idssen  7088  dflt2  10673  opsrtoslem2  16472  hausdiag  17598  hauseqlcld  17599  metustid  18474  ex-id  21590  relexpindlem  24918  dfso2  25135  dfpo2  25136  idsset  25454  dfon3  25456  elfix  25467  dffix2  25469  dffun10  25477  elfuns  25478  brsingle  25480  brapply  25501  brsuccf  25504  dfrdg4  25513  ipo0  27320  ifr0  27321
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-br 4154  df-opab 4208  df-id 4439  df-xp 4824  df-rel 4825
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