Theorem tpcoma 3785

Description: Swap 1st and 2nd members of an undordered triple. (Contributed by NM, 22-May-2015.)

Assertion

Ref	Expression
tpcoma	|- { A , B , C } = { B , A , C }

Proof of Theorem tpcoma

Step	Hyp	Ref	Expression
1		prcom 3767	. . 3 |- { A , B } = { B , A }
2	1	uneq1i 3369	. 2 |- ( { A , B } u. { C } ) = ( { B , A } u. { C } )
3		df-tp 3697	. 2 |- { A , B , C } = ( { A , B } u. { C } )
4		df-tp 3697	. 2 |- { B , A , C } = ( { B , A } u. { C } )
5	2, 3, 4	3eqtr4i 2263	1 |- { A , B , C } = { B , A , C }

Syntax hints:   = wceq 1398   u. cun 3209   {csn 3689   {cpr 3690   {ctp 3691

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-un 3215  df-pr 3696  df-tp 3697

This theorem is referenced by:  tpcomb  3786