Theorem difprsn2 3562

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 3503	. . 3 |- { A , B } = { B , A }
2	1	difeq1i 3103	. 2 |- ( { A , B } \ { B } ) = ( { B , A } \ { B } )
3		necom 2335	. . 3 |- ( A =/= B <-> B =/= A )
4		difprsn1 3561	. . 3 |- ( B =/= A -> ( { B , A } \ { B } ) = { A } )
5	3, 4	sylbi 119	. 2 |- ( A =/= B -> ( { B , A } \ { B } ) = { A } )
6	2, 5	syl5eq 2129	1 |- ( A =/= B -> ( { A , B } \ { B } ) = { A } )

Syntax hints:   -> wi 4   = wceq 1287   =/= wne 2251   \ cdif 2985   {csn 3431   {cpr 3432

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067

This theorem depends on definitions:  df-bi 115  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rab 2364  df-v 2617  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-sn 3437  df-pr 3438

This theorem is referenced by: (None)