Theorem difprsnss 3730

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 2740	. . . . 5 |- x e. _V
2	1	elpr 3613	. . . 4 |- ( x e. { A , B } <-> ( x = A \/ x = B ) )
3		velsn 3609	. . . . 5 |- ( x e. { A } <-> x = A )
4	3	notbii 668	. . . 4 |- ( -. x e. { A } <-> -. x = A )
5		biorf 744	. . . . 5 |- ( -. x = A -> ( x = B <-> ( x = A \/ x = B ) ) )
6	5	biimparc 299	. . . 4 |- ( ( ( x = A \/ x = B ) /\ -. x = A ) -> x = B )
7	2, 4, 6	syl2anb 291	. . 3 |- ( ( x e. { A , B } /\ -. x e. { A } ) -> x = B )
8		eldif 3138	. . 3 |- ( x e. ( { A , B } \ { A } ) <-> ( x e. { A , B } /\ -. x e. { A } ) )
9		velsn 3609	. . 3 |- ( x e. { B } <-> x = B )
10	7, 8, 9	3imtr4i 201	. 2 |- ( x e. ( { A , B } \ { A } ) -> x e. { B } )
11	10	ssriv 3159	1 |- ( { A , B } \ { A } ) C_ { B }

Syntax hints:   -. wn 3   /\ wa 104   \/ wo 708   = wceq 1353   e. wcel 2148   \ cdif 3126   C_ wss 3129   {csn 3592   {cpr 3593

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-sn 3598  df-pr 3599

This theorem is referenced by:  en2other2  7191