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Theorem tpidm12 3849
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 3772 . . 3  |-  { A }  =  { A ,  A }
21uneq1i 3441 . 2  |-  ( { A }  u.  { B } )  =  ( { A ,  A }  u.  { B } )
3 df-pr 3765 . 2  |-  { A ,  B }  =  ( { A }  u.  { B } )
4 df-tp 3766 . 2  |-  { A ,  A ,  B }  =  ( { A ,  A }  u.  { B } )
52, 3, 43eqtr4ri 2419 1  |-  { A ,  A ,  B }  =  { A ,  B }
Colors of variables: wff set class
Syntax hints:    = wceq 1649    u. cun 3262   {csn 3758   {cpr 3759   {ctp 3760
This theorem is referenced by:  tpidm13  3850  tpidm23  3851  tpidm  3852  hashtpg  11619
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-v 2902  df-un 3269  df-pr 3765  df-tp 3766
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