Theorem difprsnss 3846

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 2862	. . . . 5 |- x e. _V
2	1	elpr 3751	. . . 4 |- (x e. {A, B} <-> (x = A \/ x = B))
3		elsn 3748	. . . . 5 |- (x e. {A} <-> x = A)
4	3	notbii 287	. . . 4 |- (-. x e. {A} <-> -. x = A)
5		biorf 394	. . . . 5 |- (-. x = A -> (x = B <-> (x = A \/ x = B)))
6	5	biimparc 473	. . . 4 |- (((x = A \/ x = B) /\ -. x = A) -> x = B)
7	2, 4, 6	syl2anb 465	. . 3 |- ((x e. {A, B} /\ -. x e. {A}) -> x = B)
8		eldif 3221	. . 3 |- (x e. ({A, B} \ {A}) <-> (x e. {A, B} /\ -. x e. {A}))
9		elsn 3748	. . 3 |- (x e. {B} <-> x = B)
10	7, 8, 9	3imtr4i 257	. 2 |- (x e. ({A, B} \ {A}) -> x e. {B})
11	10	ssriv 3277	1 |- ({A, B} \ {A}) C_ {B}

Syntax hints:  -. wn 3   \/ wo 357   /\ wa 358   = wceq 1642   e. wcel 1710   \ cdif 3206   C_ wss 3257  {csn 3737  {cpr 3738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-sn 3741  df-pr 3742

This theorem is referenced by: (None)