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Theorem ideq 5017
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 5016 . 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 1652    e. wcel 1725   _Vcvv 2948   class class class wbr 4204    _I cid 4485
This theorem is referenced by:  dmi  5076  resieq  5148  resiexg  5180  iss  5181  imai  5210  issref  5239  intasym  5241  asymref  5242  intirr  5244  poirr2  5250  cnvi  5268  coi1  5377  dffv2  5788  idssen  7144  dflt2  10733  opsrtoslem2  16537  hausdiag  17669  hauseqlcld  17670  metustidOLD  18581  metustid  18582  ex-id  21734  relexpindlem  25131  dfso2  25369  dfpo2  25370  idsset  25727  dfon3  25729  elfix  25740  dffix2  25742  sscoid  25750  dffun10  25751  elfuns  25752  brsingle  25754  brapply  25775  brsuccf  25778  dfrdg4  25787  ipo0  27619  ifr0  27620
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877
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