Theorem tpcoma 3677

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 3659	. . 3 |- { A , B } = { B , A }
2	1	uneq1i 3277	. 2 |- ( { A , B } u. { C } ) = ( { B , A } u. { C } )
3		df-tp 3591	. 2 |- { A , B , C } = ( { A , B } u. { C } )
4		df-tp 3591	. 2 |- { B , A , C } = ( { B , A } u. { C } )
5	2, 3, 4	3eqtr4i 2201	1 |- { A , B , C } = { B , A , C }

Syntax hints:   = wceq 1348   u. cun 3119   {csn 3583   {cpr 3584   {ctp 3585

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-pr 3590  df-tp 3591

This theorem is referenced by:  tpcomb  3678