Theorem difprsn2 3526

Description: Removal of a singleton from an unordered pair. (Contributed by Alexander van der Vekens, 5-Oct-2017.)

Assertion

Ref	Expression
difprsn2	|- ( A =/= B -> ( { A , B } \ { B } ) = { A } )

Proof of Theorem difprsn2

Step	Hyp	Ref	Expression
1		prcom 3468	. . 3 |- { A , B } = { B , A }
2	1	difeq1i 3086	. 2 |- ( { A , B } \ { B } ) = ( { B , A } \ { B } )
3		necom 2329	. . 3 |- ( A =/= B <-> B =/= A )
4		difprsn1 3525	. . 3 |- ( B =/= A -> ( { B , A } \ { B } ) = { A } )
5	3, 4	sylbi 119	. 2 |- ( A =/= B -> ( { B , A } \ { B } ) = { A } )
6	2, 5	syl5eq 2125	1 |- ( A =/= B -> ( { A , B } \ { B } ) = { A } )

Syntax hints:   -> wi 4   = wceq 1284   =/= wne 2245   \ cdif 2970   {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: (None)