Theorem difprsnss 3524

Description: Removal of a singleton from an unordered pair. (Contributed by NM, 16-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Assertion

Ref	Expression
difprsnss	|- ( { A , B } \ { A } ) C_ { B }

Proof of Theorem difprsnss

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2604	. . . . 5 |- x e. _V
2	1	elpr 3419	. . . 4 |- ( x e. { A , B } <-> ( x = A \/ x = B ) )
3		velsn 3415	. . . . 5 |- ( x e. { A } <-> x = A )
4	3	notbii 626	. . . 4 |- ( -. x e. { A } <-> -. x = A )
5		biorf 695	. . . . 5 |- ( -. x = A -> ( x = B <-> ( x = A \/ x = B ) ) )
6	5	biimparc 293	. . . 4 |- ( ( ( x = A \/ x = B ) /\ -. x = A ) -> x = B )
7	2, 4, 6	syl2anb 285	. . 3 |- ( ( x e. { A , B } /\ -. x e. { A } ) -> x = B )
8		eldif 2982	. . 3 |- ( x e. ( { A , B } \ { A } ) <-> ( x e. { A , B } /\ -. x e. { A } ) )
9		velsn 3415	. . 3 |- ( x e. { B } <-> x = B )
10	7, 8, 9	3imtr4i 199	. 2 |- ( x e. ( { A , B } \ { A } ) -> x e. { B } )
11	10	ssriv 3003	1 |- ( { A , B } \ { A } ) C_ { B }

Syntax hints:   -. wn 3   /\ wa 102   \/ wo 661   = wceq 1284   e. wcel 1433   \ cdif 2970   C_ wss 2973   {csn 3398   {cpr 3399

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 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-sn 3404  df-pr 3405

This theorem is referenced by: (None)