ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  elpri GIF version

Theorem elpri 3497
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 3494 . 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 670   = wceq 1299  wcel 1448  {cpr 3475
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-un 3025  df-sn 3480  df-pr 3481
This theorem is referenced by:  nelpri  3498  nelprd  3500  opth1  4096  0nelop  4108  ontr2exmid  4378  onintexmid  4425  reg3exmidlemwe  4431  funtpg  5110  ftpg  5536  acexmidlemcase  5701  2oconcl  6266  en2eqpr  6730  eldju1st  6871  finomni  6924  exmidomniim  6925  exmidfodomrlemr  6967  exmidfodomrlemrALT  6968  sup3exmid  8573  m1expcl2  10156  maxleim  10817  maxleast  10825  zmaxcl  10835  minmax  10840  xrmaxleim  10852  xrmaxaddlem  10868  xrminmax  10873  qtopbas  12444  limcimolemlt  12513  recnprss  12529  el2oss1o  12773  nninfalllem1  12787  nninfall  12788  nninfsellemqall  12795  nninfomnilem  12798  isomninnlem  12809  trilpolemclim  12813  trilpolemcl  12814  trilpolemisumle  12815  trilpolemeq1  12817  trilpolemlt1  12818
  Copyright terms: Public domain W3C validator