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Theorem elpri 3520
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 3517 . 2 (𝐴 ∈ {𝐵, 𝐶} → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
21ibi 175 1 (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵𝐴 = 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wo 682   = wceq 1316  wcel 1465  {cpr 3498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-un 3045  df-sn 3503  df-pr 3504
This theorem is referenced by:  nelpri  3521  nelprd  3523  opth1  4128  0nelop  4140  ontr2exmid  4410  onintexmid  4457  reg3exmidlemwe  4463  funtpg  5144  ftpg  5572  acexmidlemcase  5737  2oconcl  6304  en2eqpr  6769  eldju1st  6924  finomni  6980  exmidomniim  6981  ismkvnex  6997  exmidonfinlem  7017  exmidfodomrlemr  7026  exmidfodomrlemrALT  7027  exmidaclem  7032  sup3exmid  8683  m1expcl2  10283  maxleim  10945  maxleast  10953  zmaxcl  10964  minmax  10969  xrmaxleim  10981  xrmaxaddlem  10997  xrminmax  11002  unct  11881  qtopbas  12618  limcimolemlt  12729  recnprss  12752  coseq0negpitopi  12844  el2oss1o  13115  nninfalllem1  13130  nninfall  13131  nninfsellemqall  13138  nninfomnilem  13141  isomninnlem  13152  trilpolemclim  13156  trilpolemcl  13157  trilpolemisumle  13158  trilpolemeq1  13160  trilpolemlt1  13161
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