Theorem snsslVD 39064

Description: Virtual deduction proof of snssl 39065. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
snsslVD.1	|- A e. _V

Assertion

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

Proof of Theorem snsslVD

Step	Hyp	Ref	Expression
1		idn1 38790	. . 3 |- (. { A } C_ B ->. { A } C_ B ).
2		snsslVD.1	. . . 4 |- A e. _V
3	2	snid 4208	. . 3 |- A e. { A }
4		ssel2 3598	. . 3 |- ( ( { A } C_ B /\ A e. { A } ) -> A e. B )
5	1, 3, 4	e10an 38920	. 2 |- (. { A } C_ B ->. A e. B ).
6	5	in1 38787	1 |- ( { A } C_ B -> A e. B )

Syntax hints:   -> wi 4   e. wcel 1990   _Vcvv 3200   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

This theorem is referenced by: (None)