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Theorem tpidm12 4072
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 3980 . . 3  |-  { A }  =  { A ,  A }
21uneq1i 3583 . 2  |-  ( { A }  u.  { B } )  =  ( { A ,  A }  u.  { B } )
3 df-pr 3970 . 2  |-  { A ,  B }  =  ( { A }  u.  { B } )
4 df-tp 3972 . 2  |-  { A ,  A ,  B }  =  ( { A ,  A }  u.  { B } )
52, 3, 43eqtr4ri 2483 1  |-  { A ,  A ,  B }  =  { A ,  B }
Colors of variables: wff setvar class
Syntax hints:    = wceq 1443    u. cun 3401   {csn 3967   {cpr 3969   {ctp 3971
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-v 3046  df-un 3408  df-pr 3970  df-tp 3972
This theorem is referenced by:  tpidm13  4073  tpidm23  4074  tpidm  4075  hashtpg  12638
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