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Theorem snelpwrVD 28284
Description: Virtual deduction proof of snelpwi 4350. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
snelpwrVD  |-  ( A  e.  B  ->  { A }  e.  ~P B
)

Proof of Theorem snelpwrVD
StepHypRef Expression
1 snex 4346 . . 3  |-  { A }  e.  _V
2 idn1 28006 . . . 4  |-  (. A  e.  B  ->.  A  e.  B ).
3 snssi 3885 . . . 4  |-  ( A  e.  B  ->  { A }  C_  B )
42, 3e1_ 28069 . . 3  |-  (. A  e.  B  ->.  { A }  C_  B ).
5 elpwg 3749 . . . 4  |-  ( { A }  e.  _V  ->  ( { A }  e.  ~P B  <->  { A }  C_  B ) )
65biimprd 215 . . 3  |-  ( { A }  e.  _V  ->  ( { A }  C_  B  ->  { A }  e.  ~P B
) )
71, 4, 6e01 28133 . 2  |-  (. A  e.  B  ->.  { A }  e.  ~P B ).
87in1 28003 1  |-  ( A  e.  B  ->  { A }  e.  ~P B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1717   _Vcvv 2899    C_ wss 3263   ~Pcpw 3742   {csn 3757
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-pw 3744  df-sn 3763  df-pr 3764  df-vd1 28002
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