Theorem difprsn1 4108

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 2717	. 2 |- ( B =/= A <-> A =/= B )
2		disjsn2 4035	. . . 4 |- ( B =/= A -> ( { B } i^i { A } ) = (/) )
3		disj3 3821	. . . 4 |- ( ( { B } i^i { A } ) = (/) <-> { B } = ( { B } \ { A } ) )
4	2, 3	sylib 196	. . 3 |- ( B =/= A -> { B } = ( { B } \ { A } ) )
5		df-pr 3978	. . . . . 6 |- { A , B } = ( { A } u. { B } )
6	5	equncomi 3600	. . . . 5 |- { A , B } = ( { B } u. { A } )
7	6	difeq1i 3568	. . . 4 |- ( { A , B } \ { A } ) = ( ( { B } u. { A } ) \ { A } )
8		difun2 3856	. . . 4 |- ( ( { B } u. { A } ) \ { A } ) = ( { B } \ { A } )
9	7, 8	eqtri 2480	. . 3 |- ( { A , B } \ { A } ) = ( { B } \ { A } )
10	4, 9	syl6reqr 2511	. 2 |- ( B =/= A -> ( { A , B } \ { A } ) = { B } )
11	1, 10	sylbir 213	1 |- ( A =/= B -> ( { A , B } \ { A } ) = { B } )

Syntax hints:   -> wi 4   = wceq 1370   =/= wne 2644   \ cdif 3423   u. cun 3424   i^i cin 3425   (/)c0 3735   {csn 3975   {cpr 3977

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rab 2804  df-v 3070  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-sn 3976  df-pr 3978

This theorem is referenced by:  difprsn2  4109  pmtrprfval  16095  usgra1v  23443  cusgra2v  23505  eulerpartlemgf  26896  coinflippvt  27001  f12dfv  30284  frgra2v  30729  ldepsnlinc  31157