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Theorem snsslVD 28604
Description: Virtual deduction proof of snssl 28605. (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 28342 . . 3  |-  (. { A }  C_  B  ->.  { A }  C_  B ).
2 snsslVD.1 . . . 4  |-  A  e. 
_V
32snid 3667 . . 3  |-  A  e. 
{ A }
4 ssel2 3175 . . 3  |-  ( ( { A }  C_  B  /\  A  e.  { A } )  ->  A  e.  B )
51, 3, 4e10an 28468 . 2  |-  (. { A }  C_  B  ->.  A  e.  B ).
65in1 28339 1  |-  ( { A }  C_  B  ->  A  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684   _Vcvv 2788    C_ wss 3152   {csn 3640
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-in 3159  df-ss 3166  df-sn 3646  df-vd1 28338
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