Theorem difprsn2 3758

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 3694	. . 3 |- { A , B } = { B , A }
2	1	difeq1i 3273	. 2 |- ( { A , B } \ { B } ) = ( { B , A } \ { B } )
3		necom 2448	. . 3 |- ( A =/= B <-> B =/= A )
4		difprsn1 3757	. . 3 |- ( B =/= A -> ( { B , A } \ { B } ) = { A } )
5	3, 4	sylbi 121	. 2 |- ( A =/= B -> ( { B , A } \ { B } ) = { A } )
6	2, 5	eqtrid 2238	1 |- ( A =/= B -> ( { A , B } \ { B } ) = { A } )

Syntax hints:   -> wi 4   = wceq 1364   =/= wne 2364   \ cdif 3150   {csn 3618   {cpr 3619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rab 2481  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-sn 3624  df-pr 3625

This theorem is referenced by: (None)