Theorem tpcoma 3670

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 3652	. . 3 |- { A , B } = { B , A }
2	1	uneq1i 3272	. 2 |- ( { A , B } u. { C } ) = ( { B , A } u. { C } )
3		df-tp 3584	. 2 |- { A , B , C } = ( { A , B } u. { C } )
4		df-tp 3584	. 2 |- { B , A , C } = ( { B , A } u. { C } )
5	2, 3, 4	3eqtr4i 2196	1 |- { A , B , C } = { B , A , C }

Syntax hints:   = wceq 1343   u. cun 3114   {csn 3576   {cpr 3577   {ctp 3578

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-pr 3583  df-tp 3584

This theorem is referenced by:  tpcomb  3671