Theorem tpidm12 3765

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

Step	Hyp	Ref	Expression
1		dfsn2 3680	. . 3 |- { A } = { A , A }
2	1	uneq1i 3354	. 2 |- ( { A } u. { B } ) = ( { A , A } u. { B } )
3		df-pr 3673	. 2 |- { A , B } = ( { A } u. { B } )
4		df-tp 3674	. 2 |- { A , A , B } = ( { A , A } u. { B } )
5	2, 3, 4	3eqtr4ri 2261	1 |- { A , A , B } = { A , B }

Syntax hints:   = wceq 1395   u. cun 3195   {csn 3666   {cpr 3667   {ctp 3668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-pr 3673  df-tp 3674

This theorem is referenced by:  tpidm13  3766  tpidm23  3767  tpidm  3768