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Theorem ideq 5263
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 5262 . 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 196   = wceq 1481  wcel 1988  Vcvv 3195   class class class wbr 4644   I cid 5013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-br 4645  df-opab 4704  df-id 5014  df-xp 5110  df-rel 5111
This theorem is referenced by:  dmi  5329  resieq  5395  iss  5435  restidsing  5446  restidsingOLD  5447  imai  5466  issref  5497  intasym  5499  asymref  5500  intirr  5502  poirr2  5508  cnvi  5525  xpdifid  5550  coi1  5639  dffv2  6258  isof1oidb  6559  resiexg  7087  idssen  7985  dflt2  11966  relexpindlem  13784  opsrtoslem2  19466  hausdiag  21429  hauseqlcld  21430  metustid  22340  ltgov  25473  ex-id  27261  dfso2  31619  dfpo2  31620  idsset  31972  dfon3  31974  elfix  31985  dffix2  31987  sscoid  31995  dffun10  31996  elfuns  31997  brsingle  31999  brapply  32020  brsuccf  32023  dfrdg4  32033  undmrnresiss  37729  dffrege99  38076  ipo0  38473  ifr0  38474  fourierdlem42  40129
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