Theorem difprsn1 4152

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 2723	. 2 |- ( B =/= A <-> A =/= B )
2		disjsn2 4077	. . . 4 |- ( B =/= A -> ( { B } i^i { A } ) = (/) )
3		disj3 3859	. . . 4 |- ( ( { B } i^i { A } ) = (/) <-> { B } = ( { B } \ { A } ) )
4	2, 3	sylib 196	. . 3 |- ( B =/= A -> { B } = ( { B } \ { A } ) )
5		df-pr 4019	. . . . . 6 |- { A , B } = ( { A } u. { B } )
6	5	equncomi 3636	. . . . 5 |- { A , B } = ( { B } u. { A } )
7	6	difeq1i 3604	. . . 4 |- ( { A , B } \ { A } ) = ( ( { B } u. { A } ) \ { A } )
8		difun2 3895	. . . 4 |- ( ( { B } u. { A } ) \ { A } ) = ( { B } \ { A } )
9	7, 8	eqtri 2483	. . 3 |- ( { A , B } \ { A } ) = ( { B } \ { A } )
10	4, 9	syl6reqr 2514	. 2 |- ( B =/= A -> ( { A , B } \ { A } ) = { B } )
11	1, 10	sylbir 213	1 |- ( A =/= B -> ( { A , B } \ { A } ) = { B } )

Syntax hints:   -> wi 4   = wceq 1398   =/= wne 2649   \ cdif 3458   u. cun 3459   i^i cin 3460   (/)c0 3783   {csn 4016   {cpr 4018

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-sn 4017  df-pr 4019

This theorem is referenced by:  difprsn2  4153  f12dfv  6154  pmtrprfval  16711  usgra1v  24592  cusgra2v  24664  frgra2v  25201  eulerpartlemgf  28582  coinflippvt  28687  ldepsnlinc  33363