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Theorem ideq 5101
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 5100 . 2  |-  ( B  e.  _V  ->  ( A  _I  B  <->  A  =  B ) )
31, 2ax-mp 5 1  |-  ( A  _I  B  <->  A  =  B )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1370    e. wcel 1758   _Vcvv 3078   class class class wbr 4401    _I cid 4740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-br 4402  df-opab 4460  df-id 4745  df-xp 4955  df-rel 4956
This theorem is referenced by:  dmi  5163  resieq  5230  iss  5263  restidsing  5271  imai  5290  issref  5320  intasym  5322  asymref  5323  intirr  5325  poirr2  5331  cnvi  5350  xpdifid  5375  coi1  5462  dffv2  5874  resiexg  6625  idssen  7465  dflt2  11237  opsrtoslem2  17691  hausdiag  19351  hauseqlcld  19352  metustidOLD  20267  metustid  20268  ex-id  23794  relexpindlem  27486  dfso2  27709  dfpo2  27710  idsset  28066  dfon3  28068  elfix  28079  dffix2  28081  sscoid  28089  dffun10  28090  elfuns  28091  brsingle  28093  brapply  28114  brsuccf  28117  dfrdg4  28126  ipo0  29854  ifr0  29855
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