Theorem difprsn1 3525

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 2329	. 2 |- ( B =/= A <-> A =/= B )
2		disjsn2 3455	. . . 4 |- ( B =/= A -> ( { B } i^i { A } ) = (/) )
3		disj3 3296	. . . 4 |- ( ( { B } i^i { A } ) = (/) <-> { B } = ( { B } \ { A } ) )
4	2, 3	sylib 120	. . 3 |- ( B =/= A -> { B } = ( { B } \ { A } ) )
5		df-pr 3405	. . . . . 6 |- { A , B } = ( { A } u. { B } )
6	5	equncomi 3118	. . . . 5 |- { A , B } = ( { B } u. { A } )
7	6	difeq1i 3086	. . . 4 |- ( { A , B } \ { A } ) = ( ( { B } u. { A } ) \ { A } )
8		difun2 3322	. . . 4 |- ( ( { B } u. { A } ) \ { A } ) = ( { B } \ { A } )
9	7, 8	eqtri 2101	. . 3 |- ( { A , B } \ { A } ) = ( { B } \ { A } )
10	4, 9	syl6reqr 2132	. 2 |- ( B =/= A -> ( { A , B } \ { A } ) = { B } )
11	1, 10	sylbir 133	1 |- ( A =/= B -> ( { A , B } \ { A } ) = { B } )

Syntax hints:   -> wi 4   = wceq 1284   =/= wne 2245   \ cdif 2970   u. cun 2971   i^i cin 2972   (/)c0 3251   {csn 3398   {cpr 3399

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rab 2357  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-sn 3404  df-pr 3405

This theorem is referenced by:  difprsn2  3526