Theorem difprsnss 3666

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 2692	. . . . 5 |- x e. _V
2	1	elpr 3553	. . . 4 |- ( x e. { A , B } <-> ( x = A \/ x = B ) )
3		velsn 3549	. . . . 5 |- ( x e. { A } <-> x = A )
4	3	notbii 658	. . . 4 |- ( -. x e. { A } <-> -. x = A )
5		biorf 734	. . . . 5 |- ( -. x = A -> ( x = B <-> ( x = A \/ x = B ) ) )
6	5	biimparc 297	. . . 4 |- ( ( ( x = A \/ x = B ) /\ -. x = A ) -> x = B )
7	2, 4, 6	syl2anb 289	. . 3 |- ( ( x e. { A , B } /\ -. x e. { A } ) -> x = B )
8		eldif 3085	. . 3 |- ( x e. ( { A , B } \ { A } ) <-> ( x e. { A , B } /\ -. x e. { A } ) )
9		velsn 3549	. . 3 |- ( x e. { B } <-> x = B )
10	7, 8, 9	3imtr4i 200	. 2 |- ( x e. ( { A , B } \ { A } ) -> x e. { B } )
11	10	ssriv 3106	1 |- ( { A , B } \ { A } ) C_ { B }

Syntax hints:   -. wn 3   /\ wa 103   \/ wo 698   = wceq 1332   e. wcel 1481   \ cdif 3073   C_ wss 3076   {csn 3532   {cpr 3533

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-sn 3538  df-pr 3539

This theorem is referenced by:  en2other2  7069