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Theorem snelpwrVD 27979
Description: Virtual deduction proof of snelpwi 4220. (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 4216 . . 3  |-  { A }  e.  _V
2 idn1 27715 . . . 4  |-  (. A  e.  B  ->.  A  e.  B ).
3 snssi 3759 . . . 4  |-  ( A  e.  B  ->  { A }  C_  B )
42, 3e1_ 27772 . . 3  |-  (. A  e.  B  ->.  { A }  C_  B ).
5 elpwg 3632 . . . 4  |-  ( { A }  e.  _V  ->  ( { A }  e.  ~P B  <->  { A }  C_  B ) )
65biimprd 214 . . 3  |-  ( { A }  e.  _V  ->  ( { A }  C_  B  ->  { A }  e.  ~P B
) )
71, 4, 6e01 27836 . 2  |-  (. A  e.  B  ->.  { A }  e.  ~P B ).
87in1 27712 1  |-  ( A  e.  B  ->  { A }  e.  ~P B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684   _Vcvv 2788    C_ wss 3152   ~Pcpw 3625   {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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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-ne 2448  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-pw 3627  df-sn 3646  df-pr 3647  df-vd1 27711
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