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Theorem ideq 5145
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 5144 . 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 1383    e. wcel 1804   _Vcvv 3095   class class class wbr 4437    _I cid 4780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-br 4438  df-opab 4496  df-id 4785  df-xp 4995  df-rel 4996
This theorem is referenced by:  dmi  5207  resieq  5274  iss  5311  restidsing  5320  imai  5339  issref  5370  intasym  5372  asymref  5373  intirr  5375  poirr2  5381  cnvi  5400  xpdifid  5425  coi1  5513  dffv2  5931  resiexg  6721  idssen  7562  dflt2  11364  opsrtoslem2  18127  hausdiag  20123  hauseqlcld  20124  metustidOLD  21039  metustid  21040  ltgov  23959  ex-id  25131  relexpindlem  29039  dfso2  29158  dfpo2  29159  idsset  29515  dfon3  29517  elfix  29528  dffix2  29530  sscoid  29538  dffun10  29539  elfuns  29540  brsingle  29542  brapply  29563  brsuccf  29566  dfrdg4  29575  ipo0  31312  ifr0  31313  fourierdlem42  31820
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