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Theorem ideq 5866
Description: For sets, the identity relation is the same as equality. (Contributed by NM, 13-Aug-1995.)
Hypothesis
Ref Expression
ideq.1 𝐵 ∈ V
Assertion
Ref Expression
ideq (𝐴 I 𝐵𝐴 = 𝐵)

Proof of Theorem ideq
StepHypRef Expression
1 ideq.1 . 2 𝐵 ∈ V
2 ideqg 5865 . 2 (𝐵 ∈ V → (𝐴 I 𝐵𝐴 = 𝐵))
31, 2ax-mp 5 1 (𝐴 I 𝐵𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1537  wcel 2106  Vcvv 3478   class class class wbr 5148   I cid 5582
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696
This theorem is referenced by:  dmi  5935  resieq  6011  iss  6055  elidinxp  6064  restidsing  6073  imai  6094  intasym  6138  asymref  6139  intirr  6141  poirr2  6147  cnvi  6164  xpdifid  6190  coi1  6284  dfpo2  6318  dffun2  6573  dffun2OLD  6574  dffv2  7004  isof1oidb  7344  idssen  9036  dflt2  13187  relexpindlem  15099  opsrtoslem2  22098  hausdiag  23669  hauseqlcld  23670  metustid  24583  ltgov  28620  ex-id  30463  dfso2  35735  idsset  35872  dfon3  35874  elfix  35885  dffix2  35887  sscoid  35895  dffun10  35896  elfuns  35897  brsingle  35899  brapply  35920  brsuccf  35923  dfrdg4  35933  bj-imdiridlem  37168  iss2  38326  undmrnresiss  43594  dffrege99  43952  ipo0  44445  ifr0  44446  fourierdlem42  46105
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