Theorem difprsn1 3606

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 2351	. 2 |- ( B =/= A <-> A =/= B )
2		disjsn2 3533	. . . 4 |- ( B =/= A -> ( { B } i^i { A } ) = (/) )
3		disj3 3362	. . . 4 |- ( ( { B } i^i { A } ) = (/) <-> { B } = ( { B } \ { A } ) )
4	2, 3	sylib 121	. . 3 |- ( B =/= A -> { B } = ( { B } \ { A } ) )
5		df-pr 3481	. . . . . 6 |- { A , B } = ( { A } u. { B } )
6	5	equncomi 3169	. . . . 5 |- { A , B } = ( { B } u. { A } )
7	6	difeq1i 3137	. . . 4 |- ( { A , B } \ { A } ) = ( ( { B } u. { A } ) \ { A } )
8		difun2 3389	. . . 4 |- ( ( { B } u. { A } ) \ { A } ) = ( { B } \ { A } )
9	7, 8	eqtri 2120	. . 3 |- ( { A , B } \ { A } ) = ( { B } \ { A } )
10	4, 9	syl6reqr 2151	. 2 |- ( B =/= A -> ( { A , B } \ { A } ) = { B } )
11	1, 10	sylbir 134	1 |- ( A =/= B -> ( { A , B } \ { A } ) = { B } )

Syntax hints:   -> wi 4   = wceq 1299   =/= wne 2267   \ cdif 3018   u. cun 3019   i^i cin 3020   (/)c0 3310   {csn 3474   {cpr 3475

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rab 2384  df-v 2643  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-sn 3480  df-pr 3481

This theorem is referenced by:  difprsn2  3607