Theorem snssiALTVD 33728

Description: Virtual deduction proof of snssiALT 33729. (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 3488	. . 3 |- ( { A } C_ B <-> A. x ( x e. { A } -> x e. B ) )
2		idn1 33452	. . . . . 6 |- (. A e. B ->. A e. B ).
3		idn2 33500	. . . . . . 7 |- (. A e. B ,. x e. { A } ->. x e. { A } ).
4		elsn 4046	. . . . . . 7 |- ( x e. { A } <-> x = A )
5	3, 4	e2bi 33519	. . . . . 6 |- (. A e. B ,. x e. { A } ->. x = A ).
6		eleq1a 2540	. . . . . 6 |- ( A e. B -> ( x = A -> x e. B ) )
7	2, 5, 6	e12 33622	. . . . 5 |- (. A e. B ,. x e. { A } ->. x e. B ).
8	7	in2 33492	. . . 4 |- (. A e. B ->. ( x e. { A } -> x e. B ) ).
9	8	gen11 33503	. . 3 |- (. A e. B ->. A. x ( x e. { A } -> x e. B ) ).
10		bi2 198	. . 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 33578	. 2 |- (. A e. B ->. { A } C_ B ).
12	11	in1 33449	1 |- ( A e. B -> { A } C_ B )

Syntax hints:   -> wi 4   <-> wb 184   A.wal 1393   = wceq 1395   e. wcel 1819   C_ wss 3471   {csn 4032

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-in 3478  df-ss 3485  df-sn 4033  df-vd1 33448  df-vd2 33456

This theorem is referenced by: (None)