Theorem difprsnss 3658

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 2689	. . . . 5 |- x e. _V
2	1	elpr 3548	. . . 4 |- ( x e. { A , B } <-> ( x = A \/ x = B ) )
3		velsn 3544	. . . . 5 |- ( x e. { A } <-> x = A )
4	3	notbii 657	. . . 4 |- ( -. x e. { A } <-> -. x = A )
5		biorf 733	. . . . 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 3080	. . 3 |- ( x e. ( { A , B } \ { A } ) <-> ( x e. { A , B } /\ -. x e. { A } ) )
9		velsn 3544	. . 3 |- ( x e. { B } <-> x = B )
10	7, 8, 9	3imtr4i 200	. 2 |- ( x e. ( { A , B } \ { A } ) -> x e. { B } )
11	10	ssriv 3101	1 |- ( { A , B } \ { A } ) C_ { B }

Syntax hints:   -. wn 3   /\ wa 103   \/ wo 697   = wceq 1331   e. wcel 1480   \ cdif 3068   C_ wss 3071   {csn 3527   {cpr 3528

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-pr 3534

This theorem is referenced by:  en2other2  7052