Theorem difprsn2 3720

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 3659	. . 3 |- { A , B } = { B , A }
2	1	difeq1i 3241	. 2 |- ( { A , B } \ { B } ) = ( { B , A } \ { B } )
3		necom 2424	. . 3 |- ( A =/= B <-> B =/= A )
4		difprsn1 3719	. . 3 |- ( B =/= A -> ( { B , A } \ { B } ) = { A } )
5	3, 4	sylbi 120	. 2 |- ( A =/= B -> ( { B , A } \ { B } ) = { A } )
6	2, 5	eqtrid 2215	1 |- ( A =/= B -> ( { A , B } \ { B } ) = { A } )

Syntax hints:   -> wi 4   = wceq 1348   =/= wne 2340   \ cdif 3118   {csn 3583   {cpr 3584

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-sn 3589  df-pr 3590

This theorem is referenced by: (None)