Theorem tpcoma 3704

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 3686	. . 3 |- { A , B } = { B , A }
2	1	uneq1i 3300	. 2 |- ( { A , B } u. { C } ) = ( { B , A } u. { C } )
3		df-tp 3618	. 2 |- { A , B , C } = ( { A , B } u. { C } )
4		df-tp 3618	. 2 |- { B , A , C } = ( { B , A } u. { C } )
5	2, 3, 4	3eqtr4i 2220	1 |- { A , B , C } = { B , A , C }

Syntax hints:   = wceq 1364   u. cun 3142   {csn 3610   {cpr 3611   {ctp 3612

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-pr 3617  df-tp 3618

This theorem is referenced by:  tpcomb  3705