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Theorem tpidm12 4128
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 4040 . . 3  |-  { A }  =  { A ,  A }
21uneq1i 3654 . 2  |-  ( { A }  u.  { B } )  =  ( { A ,  A }  u.  { B } )
3 df-pr 4030 . 2  |-  { A ,  B }  =  ( { A }  u.  { B } )
4 df-tp 4032 . 2  |-  { A ,  A ,  B }  =  ( { A ,  A }  u.  { B } )
52, 3, 43eqtr4ri 2507 1  |-  { A ,  A ,  B }  =  { A ,  B }
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    u. cun 3474   {csn 4027   {cpr 4029   {ctp 4031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3115  df-un 3481  df-pr 4030  df-tp 4032
This theorem is referenced by:  tpidm13  4129  tpidm23  4130  tpidm  4131  hashtpg  12490
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