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Theorem tpidm12 3897
Description: Unordered triple  { A ,  A ,  B } is just an overlong way to write  { A ,  B }. (Contributed by David A. Wheeler, 10-May-2015.)
Assertion
Ref Expression
tpidm12  |-  { A ,  A ,  B }  =  { A ,  B }

Proof of Theorem tpidm12
StepHypRef Expression
1 dfsn2 3820 . . 3  |-  { A }  =  { A ,  A }
21uneq1i 3489 . 2  |-  ( { A }  u.  { B } )  =  ( { A ,  A }  u.  { B } )
3 df-pr 3813 . 2  |-  { A ,  B }  =  ( { A }  u.  { B } )
4 df-tp 3814 . 2  |-  { A ,  A ,  B }  =  ( { A ,  A }  u.  { B } )
52, 3, 43eqtr4ri 2466 1  |-  { A ,  A ,  B }  =  { A ,  B }
Colors of variables: wff set class
Syntax hints:    = wceq 1652    u. cun 3310   {csn 3806   {cpr 3807   {ctp 3808
This theorem is referenced by:  tpidm13  3898  tpidm23  3899  tpidm  3900  hashtpg  11683
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-un 3317  df-pr 3813  df-tp 3814
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