Theorem snssiALTVD 39062

Description: Virtual deduction proof of snssiALT 39063. (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 3591	. . 3 |- ( { A } C_ B <-> A. x ( x e. { A } -> x e. B ) )
2		idn1 38790	. . . . . 6 |- (. A e. B ->. A e. B ).
3		idn2 38838	. . . . . . 7 |- (. A e. B ,. x e. { A } ->. x e. { A } ).
4		velsn 4193	. . . . . . 7 |- ( x e. { A } <-> x = A )
5	3, 4	e2bi 38857	. . . . . 6 |- (. A e. B ,. x e. { A } ->. x = A ).
6		eleq1a 2696	. . . . . 6 |- ( A e. B -> ( x = A -> x e. B ) )
7	2, 5, 6	e12 38951	. . . . 5 |- (. A e. B ,. x e. { A } ->. x e. B ).
8	7	in2 38830	. . . 4 |- (. A e. B ->. ( x e. { A } -> x e. B ) ).
9	8	gen11 38841	. . 3 |- (. A e. B ->. A. x ( x e. { A } -> x e. B ) ).
10		biimpr 210	. . 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 38916	. 2 |- (. A e. B ->. { A } C_ B ).
12	11	in1 38787	1 |- ( A e. B -> { A } C_ B )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   = wceq 1483   e. wcel 1990   C_ wss 3574   {csn 4177

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-in 3581  df-ss 3588  df-sn 4178  df-vd1 38786  df-vd2 38794

This theorem is referenced by: (None)