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Theorem tpidm12 3741
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 3667 . . 3  |-  { A }  =  { A ,  A }
21uneq1i 3338 . 2  |-  ( { A }  u.  { B } )  =  ( { A ,  A }  u.  { B } )
3 df-pr 3660 . 2  |-  { A ,  B }  =  ( { A }  u.  { B } )
4 df-tp 3661 . 2  |-  { A ,  A ,  B }  =  ( { A ,  A }  u.  { B } )
52, 3, 43eqtr4ri 2327 1  |-  { A ,  A ,  B }  =  { A ,  B }
Colors of variables: wff set class
Syntax hints:    = wceq 1632    u. cun 3163   {csn 3653   {cpr 3654   {ctp 3655
This theorem is referenced by:  tpidm13  3742  tpidm23  3743  tpidm  3744  hashtpg  28217
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-un 3170  df-pr 3660  df-tp 3661
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