Theorem tpcoma 3726

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 3708	. . 3 |- { A , B } = { B , A }
2	1	uneq1i 3322	. 2 |- ( { A , B } u. { C } ) = ( { B , A } u. { C } )
3		df-tp 3640	. 2 |- { A , B , C } = ( { A , B } u. { C } )
4		df-tp 3640	. 2 |- { B , A , C } = ( { B , A } u. { C } )
5	2, 3, 4	3eqtr4i 2235	1 |- { A , B , C } = { B , A , C }

Syntax hints:   = wceq 1372   u. cun 3163   {csn 3632   {cpr 3633   {ctp 3634

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-un 3169  df-pr 3639  df-tp 3640

This theorem is referenced by:  tpcomb  3727