Theorem difprsn2 3495

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 3437	. . 3 |- {A , B } = {B , A }
2	1	difeq1i 3052	. 2 |- ( {A , B } \ {B }) = ( {B , A } \ {B })
3		necom 2283	. . 3 |- (A =/= B <-> B =/= A)
4		difprsn1 3494	. . 3 |- (B =/= A -> ( {B , A } \ {B }) = {A })
5	3, 4	sylbi 114	. 2 |- (A =/= B -> ( {B , A } \ {B }) = {A })
6	2, 5	syl5eq 2081	1 |- (A =/= B -> ( {A , B } \ {B }) = {A })

Syntax hints:   -> wi 4   = wceq 1242   =/= wne 2201   \ cdif 2908   {csn 3367   {cpr 3368

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-sn 3373  df-pr 3374

This theorem is referenced by: (None)