Theorem difprsnss 3711

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 2729	. . . . 5 |- x e. _V
2	1	elpr 3597	. . . 4 |- ( x e. { A , B } <-> ( x = A \/ x = B ) )
3		velsn 3593	. . . . 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 3125	. . 3 |- ( x e. ( { A , B } \ { A } ) <-> ( x e. { A , B } /\ -. x e. { A } ) )
9		velsn 3593	. . 3 |- ( x e. { B } <-> x = B )
10	7, 8, 9	3imtr4i 200	. 2 |- ( x e. ( { A , B } \ { A } ) -> x e. { B } )
11	10	ssriv 3146	1 |- ( { A , B } \ { A } ) C_ { B }

Syntax hints:   -. wn 3   /\ wa 103   \/ wo 698   = wceq 1343   e. wcel 2136   \ cdif 3113   C_ wss 3116   {csn 3576   {cpr 3577

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-sn 3582  df-pr 3583

This theorem is referenced by:  en2other2  7152