Theorem difprsn1 3812

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 2486	. 2 |- ( B =/= A <-> A =/= B )
2		df-pr 3676	. . . . . 6 |- { A , B } = ( { A } u. { B } )
3	2	equncomi 3353	. . . . 5 |- { A , B } = ( { B } u. { A } )
4	3	difeq1i 3321	. . . 4 |- ( { A , B } \ { A } ) = ( ( { B } u. { A } ) \ { A } )
5		difun2 3574	. . . 4 |- ( ( { B } u. { A } ) \ { A } ) = ( { B } \ { A } )
6	4, 5	eqtri 2252	. . 3 |- ( { A , B } \ { A } ) = ( { B } \ { A } )
7		disjsn2 3732	. . . 4 |- ( B =/= A -> ( { B } i^i { A } ) = (/) )
8		disj3 3547	. . . 4 |- ( ( { B } i^i { A } ) = (/) <-> { B } = ( { B } \ { A } ) )
9	7, 8	sylib 122	. . 3 |- ( B =/= A -> { B } = ( { B } \ { A } ) )
10	6, 9	eqtr4id 2283	. 2 |- ( B =/= A -> ( { A , B } \ { A } ) = { B } )
11	1, 10	sylbir 135	1 |- ( A =/= B -> ( { A , B } \ { A } ) = { B } )

Syntax hints:   -> wi 4   = wceq 1397   =/= wne 2402   \ cdif 3197   u. cun 3198   i^i cin 3199   (/)c0 3494   {csn 3669   {cpr 3670

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rab 2519  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-sn 3675  df-pr 3676

This theorem is referenced by:  difprsn2  3813