Theorem tpcoma 3801

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 3783	. . 3 |- { A , B } = { B , A }
2	1	uneq1i 3379	. 2 |- ( { A , B } u. { C } ) = ( { B , A } u. { C } )
3		df-tp 3713	. 2 |- { A , B , C } = ( { A , B } u. { C } )
4		df-tp 3713	. 2 |- { B , A , C } = ( { B , A } u. { C } )
5	2, 3, 4	3eqtr4i 2269	1 |- { A , B , C } = { B , A , C }

Syntax hints:   = wceq 1402   u. cun 3218   {csn 3705   {cpr 3706   {ctp 3707

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-pr 3712  df-tp 3713

This theorem is referenced by:  tpcomb  3802