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Theorem snsslVD 28941
Description: Virtual deduction proof of snssl 28942. (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 28665 . . 3  |-  (. { A }  C_  B  ->.  { A }  C_  B ).
2 snsslVD.1 . . . 4  |-  A  e. 
_V
32snid 3841 . . 3  |-  A  e. 
{ A }
4 ssel2 3343 . . 3  |-  ( ( { A }  C_  B  /\  A  e.  { A } )  ->  A  e.  B )
51, 3, 4e10an 28796 . 2  |-  (. { A }  C_  B  ->.  A  e.  B ).
65in1 28662 1  |-  ( { A }  C_  B  ->  A  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725   _Vcvv 2956    C_ wss 3320   {csn 3814
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-in 3327  df-ss 3334  df-sn 3820  df-vd1 28661
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