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Theorem ideq 5146
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 5145 . 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 1374    e. wcel 1762   _Vcvv 3106   class class class wbr 4440    _I cid 4783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-br 4441  df-opab 4499  df-id 4788  df-xp 4998  df-rel 4999
This theorem is referenced by:  dmi  5208  resieq  5275  iss  5312  restidsing  5321  imai  5340  issref  5371  intasym  5373  asymref  5374  intirr  5376  poirr2  5382  cnvi  5401  xpdifid  5426  coi1  5514  dffv2  5931  resiexg  6710  idssen  7550  dflt2  11343  opsrtoslem2  17913  hausdiag  19874  hauseqlcld  19875  metustidOLD  20790  metustid  20791  ltgov  23703  ex-id  24818  relexpindlem  28523  dfso2  28746  dfpo2  28747  idsset  29103  dfon3  29105  elfix  29116  dffix2  29118  sscoid  29126  dffun10  29127  elfuns  29128  brsingle  29130  brapply  29151  brsuccf  29154  dfrdg4  29163  ipo0  30891  ifr0  30892
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