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Theorem snsslVD 31847
Description: Virtual deduction proof of snssl 31848. (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 31569 . . 3  |-  (. { A }  C_  B  ->.  { A }  C_  B ).
2 snsslVD.1 . . . 4  |-  A  e. 
_V
32snid 3989 . . 3  |-  A  e. 
{ A }
4 ssel2 3435 . . 3  |-  ( ( { A }  C_  B  /\  A  e.  { A } )  ->  A  e.  B )
51, 3, 4e10an 31699 . 2  |-  (. { A }  C_  B  ->.  A  e.  B ).
65in1 31566 1  |-  ( { A }  C_  B  ->  A  e.  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1757   _Vcvv 3054    C_ wss 3412   {csn 3961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-v 3056  df-in 3419  df-ss 3426  df-sn 3962  df-vd1 31565
This theorem is referenced by: (None)
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