Theorem difprsn1 3806

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 2484	. 2 |- ( B =/= A <-> A =/= B )
2		df-pr 3673	. . . . . 6 |- { A , B } = ( { A } u. { B } )
3	2	equncomi 3350	. . . . 5 |- { A , B } = ( { B } u. { A } )
4	3	difeq1i 3318	. . . 4 |- ( { A , B } \ { A } ) = ( ( { B } u. { A } ) \ { A } )
5		difun2 3571	. . . 4 |- ( ( { B } u. { A } ) \ { A } ) = ( { B } \ { A } )
6	4, 5	eqtri 2250	. . 3 |- ( { A , B } \ { A } ) = ( { B } \ { A } )
7		disjsn2 3729	. . . 4 |- ( B =/= A -> ( { B } i^i { A } ) = (/) )
8		disj3 3544	. . . 4 |- ( ( { B } i^i { A } ) = (/) <-> { B } = ( { B } \ { A } ) )
9	7, 8	sylib 122	. . 3 |- ( B =/= A -> { B } = ( { B } \ { A } ) )
10	6, 9	eqtr4id 2281	. 2 |- ( B =/= A -> ( { A , B } \ { A } ) = { B } )
11	1, 10	sylbir 135	1 |- ( A =/= B -> ( { A , B } \ { A } ) = { B } )

Syntax hints:   -> wi 4   = wceq 1395   =/= wne 2400   \ cdif 3194   u. cun 3195   i^i cin 3196   (/)c0 3491   {csn 3666   {cpr 3667

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rab 2517  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-sn 3672  df-pr 3673

This theorem is referenced by:  difprsn2  3807