Theorem difprsnss 4329

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 3203	. . . . 5 |- x e. _V
2	1	elpr 4198	. . . 4 |- ( x e. { A , B } <-> ( x = A \/ x = B ) )
3		velsn 4193	. . . . 5 |- ( x e. { A } <-> x = A )
4	3	notbii 310	. . . 4 |- ( -. x e. { A } <-> -. x = A )
5		biorf 420	. . . . 5 |- ( -. x = A -> ( x = B <-> ( x = A \/ x = B ) ) )
6	5	biimparc 504	. . . 4 |- ( ( ( x = A \/ x = B ) /\ -. x = A ) -> x = B )
7	2, 4, 6	syl2anb 496	. . 3 |- ( ( x e. { A , B } /\ -. x e. { A } ) -> x = B )
8		eldif 3584	. . 3 |- ( x e. ( { A , B } \ { A } ) <-> ( x e. { A , B } /\ -. x e. { A } ) )
9		velsn 4193	. . 3 |- ( x e. { B } <-> x = B )
10	7, 8, 9	3imtr4i 281	. 2 |- ( x e. ( { A , B } \ { A } ) -> x e. { B } )
11	10	ssriv 3607	1 |- ( { A , B } \ { A } ) C_ { B }

Syntax hints:   -. wn 3   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   \ cdif 3571   C_ wss 3574   {csn 4177   {cpr 4179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-sn 4178  df-pr 4180

This theorem is referenced by:  en2other2  8832  pmtrprfv  17873  itg11  23458