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Theorem ideq 5862
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 5861 . 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 1539  wcel 2107  Vcvv 3479   class class class wbr 5142   I cid 5576
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691
This theorem is referenced by:  dmi  5931  resieq  6007  iss  6052  elidinxp  6061  restidsing  6070  imai  6091  intasym  6134  asymref  6135  intirr  6137  poirr2  6143  cnvi  6160  xpdifid  6187  coi1  6281  dfpo2  6315  dffun2  6570  dffun2OLD  6571  dffv2  7003  isof1oidb  7345  idssen  9038  dflt2  13191  relexpindlem  15103  opsrtoslem2  22081  hausdiag  23654  hauseqlcld  23655  metustid  24568  ltgov  28606  ex-id  30454  dfso2  35756  idsset  35892  dfon3  35894  elfix  35905  dffix2  35907  sscoid  35915  dffun10  35916  elfuns  35917  brsingle  35919  brapply  35940  brsuccf  35943  dfrdg4  35953  bj-imdiridlem  37187  iss2  38346  undmrnresiss  43622  dffrege99  43980  ipo0  44473  ifr0  44474  fourierdlem42  46169
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