Theorem difprsn2 4331

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 4267	. . 3 |- { A , B } = { B , A }
2	1	difeq1i 3724	. 2 |- ( { A , B } \ { B } ) = ( { B , A } \ { B } )
3		necom 2847	. . 3 |- ( A =/= B <-> B =/= A )
4		difprsn1 4330	. . 3 |- ( B =/= A -> ( { B , A } \ { B } ) = { A } )
5	3, 4	sylbi 207	. 2 |- ( A =/= B -> ( { B , A } \ { B } ) = { A } )
6	2, 5	syl5eq 2668	1 |- ( A =/= B -> ( { A , B } \ { B } ) = { A } )

Syntax hints:   -> wi 4   = wceq 1483   =/= wne 2794   \ cdif 3571   {csn 4177   {cpr 4179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-pr 4180

This theorem is referenced by:  f12dfv  6529  pmtrprfval  17907  nbgr2vtx1edg  26246  nbuhgr2vtx1edgb  26248  nfrgr2v  27136  ldepsnlinc  42297