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Theorem elpri 3464
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 3461 . 2 (𝐴 ∈ {𝐵, 𝐶} → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
21ibi 174 1 (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵𝐴 = 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wo 664   = wceq 1289  wcel 1438  {cpr 3442
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-sn 3447  df-pr 3448
This theorem is referenced by:  nelpri  3465  nelprd  3467  opth1  4054  0nelop  4066  ontr2exmid  4331  onintexmid  4378  reg3exmidlemwe  4384  funtpg  5051  ftpg  5465  acexmidlemcase  5629  2oconcl  6185  en2eqpr  6603  eldju1st  6741  finomni  6775  exmidomniim  6776  exmidfodomrlemr  6807  exmidfodomrlemrALT  6808  m1expcl2  9942  maxleim  10603  maxleast  10611  zmaxcl  10621  minmax  10625  el2oss1o  11544  nninfalllem1  11556  nninfall  11557  nninfsellemqall  11564  nninfomnilem  11567
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