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Theorem tpidm12 4085
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 3999 . . 3  |-  { A }  =  { A ,  A }
21uneq1i 3615 . 2  |-  ( { A }  u.  { B } )  =  ( { A ,  A }  u.  { B } )
3 df-pr 3989 . 2  |-  { A ,  B }  =  ( { A }  u.  { B } )
4 df-tp 3991 . 2  |-  { A ,  A ,  B }  =  ( { A ,  A }  u.  { B } )
52, 3, 43eqtr4ri 2494 1  |-  { A ,  A ,  B }  =  { A ,  B }
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370    u. cun 3435   {csn 3986   {cpr 3988   {ctp 3990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3080  df-un 3442  df-pr 3989  df-tp 3991
This theorem is referenced by:  tpidm13  4086  tpidm23  4087  tpidm  4088  hashtpg  12305
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