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Theorem tpidm12 3905
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 3828 . . 3  |-  { A }  =  { A ,  A }
21uneq1i 3497 . 2  |-  ( { A }  u.  { B } )  =  ( { A ,  A }  u.  { B } )
3 df-pr 3821 . 2  |-  { A ,  B }  =  ( { A }  u.  { B } )
4 df-tp 3822 . 2  |-  { A ,  A ,  B }  =  ( { A ,  A }  u.  { B } )
52, 3, 43eqtr4ri 2467 1  |-  { A ,  A ,  B }  =  { A ,  B }
Colors of variables: wff set class
Syntax hints:    = wceq 1652    u. cun 3318   {csn 3814   {cpr 3815   {ctp 3816
This theorem is referenced by:  tpidm13  3906  tpidm23  3907  tpidm  3908  hashtpg  11691
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 2417
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 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-un 3325  df-pr 3821  df-tp 3822
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