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Theorem elpri 3630
Description: If a class is an element of a pair, then it is one of the two paired elements. (Contributed by Scott Fenton, 1-Apr-2011.)
Assertion
Ref Expression
elpri (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵𝐴 = 𝐶))

Proof of Theorem elpri
StepHypRef Expression
1 elprg 3627 . 2 (𝐴 ∈ {𝐵, 𝐶} → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
21ibi 176 1 (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵𝐴 = 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wo 709   = wceq 1364  wcel 2160  {cpr 3608
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614
This theorem is referenced by:  nelpri  3631  nelprd  3633  opth1  4254  0nelop  4266  ontr2exmid  4542  onintexmid  4590  reg3exmidlemwe  4596  funtpg  5286  ftpg  5721  acexmidlemcase  5892  2oconcl  6465  el2oss1o  6469  pw2f1odclem  6863  en2eqpr  6936  eldju1st  7101  nninfisol  7162  finomni  7169  exmidomniim  7170  ismkvnex  7184  nninfwlpoimlemginf  7205  exmidonfinlem  7223  exmidfodomrlemr  7232  exmidfodomrlemrALT  7233  exmidaclem  7238  sup3exmid  8945  m1expcl2  10576  maxleim  11249  maxleast  11257  zmaxcl  11268  minmax  11273  xrmaxleim  11287  xrmaxaddlem  11303  xrminmax  11308  prm23lt5  12298  unct  12496  fnpr2ob  12819  fvprif  12822  xpsfeq  12824  qtopbas  14499  limcimolemlt  14610  recnprss  14633  coseq0negpitopi  14734  lgslem4  14882  lgseisenlem2  14929  2lgsoddprmlem3  14937  012of  15224  2o01f  15225  nninfalllem1  15236  nninfall  15237  nninfsellemqall  15243  nninfomnilem  15246  trilpolemclim  15263  trilpolemcl  15264  trilpolemisumle  15265  trilpolemeq1  15267  trilpolemlt1  15268  iswomni0  15278  nconstwlpolemgt0  15291  nconstwlpolem  15292
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