ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  tppreq3 Unicode version

Theorem tppreq3 3540
Description: An unordered triple is an unordered pair if one of its elements is identical with another element. (Contributed by Alexander van der Vekens, 6-Oct-2017.)
Assertion
Ref Expression
tppreq3  |-  ( B  =  C  ->  { A ,  B ,  C }  =  { A ,  B } )

Proof of Theorem tppreq3
StepHypRef Expression
1 tpeq3 3525 . . 3  |-  ( C  =  B  ->  { A ,  B ,  C }  =  { A ,  B ,  B } )
21eqcoms 2091 . 2  |-  ( B  =  C  ->  { A ,  B ,  C }  =  { A ,  B ,  B } )
3 tpidm23 3538 . 2  |-  { A ,  B ,  B }  =  { A ,  B }
42, 3syl6eq 2136 1  |-  ( B  =  C  ->  { A ,  B ,  C }  =  { A ,  B } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289   {cpr 3442   {ctp 3443
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-3or 925  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  df-tp 3449
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator