Theorem difprsn1 3771

Description: Removal of a singleton from an unordered pair. (Contributed by Thierry Arnoux, 4-Feb-2017.)

Assertion

Ref	Expression
difprsn1	|- ( A =/= B -> ( { A , B } \ { A } ) = { B } )

Proof of Theorem difprsn1

Step	Hyp	Ref	Expression
1		necom 2459	. 2 |- ( B =/= A <-> A =/= B )
2		df-pr 3639	. . . . . 6 |- { A , B } = ( { A } u. { B } )
3	2	equncomi 3318	. . . . 5 |- { A , B } = ( { B } u. { A } )
4	3	difeq1i 3286	. . . 4 |- ( { A , B } \ { A } ) = ( ( { B } u. { A } ) \ { A } )
5		difun2 3539	. . . 4 |- ( ( { B } u. { A } ) \ { A } ) = ( { B } \ { A } )
6	4, 5	eqtri 2225	. . 3 |- ( { A , B } \ { A } ) = ( { B } \ { A } )
7		disjsn2 3695	. . . 4 |- ( B =/= A -> ( { B } i^i { A } ) = (/) )
8		disj3 3512	. . . 4 |- ( ( { B } i^i { A } ) = (/) <-> { B } = ( { B } \ { A } ) )
9	7, 8	sylib 122	. . 3 |- ( B =/= A -> { B } = ( { B } \ { A } ) )
10	6, 9	eqtr4id 2256	. 2 |- ( B =/= A -> ( { A , B } \ { A } ) = { B } )
11	1, 10	sylbir 135	1 |- ( A =/= B -> ( { A , B } \ { A } ) = { B } )

Syntax hints:   -> wi 4   = wceq 1372   =/= wne 2375   \ cdif 3162   u. cun 3163   i^i cin 3164   (/)c0 3459   {csn 3632   {cpr 3633

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-in1 615  ax-in2 616  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-fal 1378  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rab 2492  df-v 2773  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-sn 3638  df-pr 3639

This theorem is referenced by:  difprsn2  3772