Theorem difprsn1 3503

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 2289	. 2 |- ( B =/= A <-> A =/= B )
2		disjsn2 3433	. . . 4 |- ( B =/= A -> ( { B } i^i { A } ) = (/) )
3		disj3 3272	. . . 4 |- ( ( { B } i^i { A } ) = (/) <-> { B } = ( { B } \ { A } ) )
4	2, 3	sylib 127	. . 3 |- ( B =/= A -> { B } = ( { B } \ { A } ) )
5		df-pr 3382	. . . . . 6 |- { A , B } = ( { A } u. { B } )
6	5	equncomi 3089	. . . . 5 |- { A , B } = ( { B } u. { A } )
7	6	difeq1i 3058	. . . 4 |- ( { A , B } \ { A } ) = ( ( { B } u. { A } ) \ { A } )
8		difun2 3302	. . . 4 |- ( ( { B } u. { A } ) \ { A } ) = ( { B } \ { A } )
9	7, 8	eqtri 2060	. . 3 |- ( { A , B } \ { A } ) = ( { B } \ { A } )
10	4, 9	syl6reqr 2091	. 2 |- ( B =/= A -> ( { A , B } \ { A } ) = { B } )
11	1, 10	sylbir 125	1 |- ( A =/= B -> ( { A , B } \ { A } ) = { B } )

Syntax hints:   -> wi 4   = wceq 1243   =/= wne 2204   \ cdif 2914   u. cun 2915   i^i cin 2916   (/)c0 3224   {csn 3375   {cpr 3376

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-sn 3381  df-pr 3382

This theorem is referenced by:  difprsn2  3504