Theorem tpidm12 3732

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 3647	. . 3 |- { A } = { A , A }
2	1	uneq1i 3323	. 2 |- ( { A } u. { B } ) = ( { A , A } u. { B } )
3		df-pr 3640	. 2 |- { A , B } = ( { A } u. { B } )
4		df-tp 3641	. 2 |- { A , A , B } = ( { A , A } u. { B } )
5	2, 3, 4	3eqtr4ri 2237	1 |- { A , A , B } = { A , B }

Syntax hints:   = wceq 1373   u. cun 3164   {csn 3633   {cpr 3634   {ctp 3635

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-pr 3640  df-tp 3641

This theorem is referenced by:  tpidm13  3733  tpidm23  3734  tpidm  3735