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Theorem tpidm12 4095
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 4006 . . 3  |-  { A }  =  { A ,  A }
21uneq1i 3613 . 2  |-  ( { A }  u.  { B } )  =  ( { A ,  A }  u.  { B } )
3 df-pr 3996 . 2  |-  { A ,  B }  =  ( { A }  u.  { B } )
4 df-tp 3998 . 2  |-  { A ,  A ,  B }  =  ( { A ,  A }  u.  { B } )
52, 3, 43eqtr4ri 2460 1  |-  { A ,  A ,  B }  =  { A ,  B }
Colors of variables: wff setvar class
Syntax hints:    = wceq 1437    u. cun 3431   {csn 3993   {cpr 3995   {ctp 3997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-v 3080  df-un 3438  df-pr 3996  df-tp 3998
This theorem is referenced by:  tpidm13  4096  tpidm23  4097  tpidm  4098  hashtpg  12621
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