Theorem difprsn2 3713

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 3652	. . 3 |- { A , B } = { B , A }
2	1	difeq1i 3236	. 2 |- ( { A , B } \ { B } ) = ( { B , A } \ { B } )
3		necom 2420	. . 3 |- ( A =/= B <-> B =/= A )
4		difprsn1 3712	. . 3 |- ( B =/= A -> ( { B , A } \ { B } ) = { A } )
5	3, 4	sylbi 120	. 2 |- ( A =/= B -> ( { B , A } \ { B } ) = { A } )
6	2, 5	syl5eq 2211	1 |- ( A =/= B -> ( { A , B } \ { B } ) = { A } )

Syntax hints:   -> wi 4   = wceq 1343   =/= wne 2336   \ cdif 3113   {csn 3576   {cpr 3577

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rab 2453  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-sn 3582  df-pr 3583

This theorem is referenced by: (None)