Theorem snssiALTVD 37223

Description: Virtual deduction proof of snssiALT 37224. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
snssiALTVD	|- ( A e. B -> { A } C_ B )

Proof of Theorem snssiALTVD

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfss2 3421	. . 3 |- ( { A } C_ B <-> A. x ( x e. { A } -> x e. B ) )
2		idn1 36944	. . . . . 6 |- (. A e. B ->. A e. B ).
3		idn2 36992	. . . . . . 7 |- (. A e. B ,. x e. { A } ->. x e. { A } ).
4		elsn 3982	. . . . . . 7 |- ( x e. { A } <-> x = A )
5	3, 4	e2bi 37011	. . . . . 6 |- (. A e. B ,. x e. { A } ->. x = A ).
6		eleq1a 2524	. . . . . 6 |- ( A e. B -> ( x = A -> x e. B ) )
7	2, 5, 6	e12 37111	. . . . 5 |- (. A e. B ,. x e. { A } ->. x e. B ).
8	7	in2 36984	. . . 4 |- (. A e. B ->. ( x e. { A } -> x e. B ) ).
9	8	gen11 36995	. . 3 |- (. A e. B ->. A. x ( x e. { A } -> x e. B ) ).
10		biimpr 202	. . 3 |- ( ( { A } C_ B <-> A. x ( x e. { A } -> x e. B ) ) -> ( A. x ( x e. { A } -> x e. B ) -> { A } C_ B ) )
11	1, 9, 10	e01 37070	. 2 |- (. A e. B ->. { A } C_ B ).
12	11	in1 36941	1 |- ( A e. B -> { A } C_ B )

Syntax hints:   -> wi 4   <-> wb 188   A.wal 1442   = wceq 1444   e. wcel 1887   C_ wss 3404   {csn 3968

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431

This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-in 3411  df-ss 3418  df-sn 3969  df-vd1 36940  df-vd2 36948

This theorem is referenced by: (None)