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Theorem snsslVD 28920
Description: Virtual deduction proof of snssl 28921. (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
StepHypRef Expression
1 idn1 28641 . . 3  |-  (. { A }  C_  B  ->.  { A }  C_  B ).
2 snsslVD.1 . . . 4  |-  A  e. 
_V
32snid 3680 . . 3  |-  A  e. 
{ A }
4 ssel2 3188 . . 3  |-  ( ( { A }  C_  B  /\  A  e.  { A } )  ->  A  e.  B )
51, 3, 4e10an 28773 . 2  |-  (. { A }  C_  B  ->.  A  e.  B ).
65in1 28638 1  |-  ( { A }  C_  B  ->  A  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1696   _Vcvv 2801    C_ wss 3165   {csn 3653
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-in 3172  df-ss 3179  df-sn 3659  df-vd1 28637
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