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Theorem snelpwrVD 33112
Description: Virtual deduction proof of snelpwi 4698. (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 4694 . . 3  |-  { A }  e.  _V
2 idn1 32832 . . . 4  |-  (. A  e.  B  ->.  A  e.  B ).
3 snssi 4177 . . . 4  |-  ( A  e.  B  ->  { A }  C_  B )
42, 3e1a 32894 . . 3  |-  (. A  e.  B  ->.  { A }  C_  B ).
5 elpwg 4024 . . . 4  |-  ( { A }  e.  _V  ->  ( { A }  e.  ~P B  <->  { A }  C_  B ) )
65biimprd 223 . . 3  |-  ( { A }  e.  _V  ->  ( { A }  C_  B  ->  { A }  e.  ~P B
) )
71, 4, 6e01 32958 . 2  |-  (. A  e.  B  ->.  { A }  e.  ~P B ).
87in1 32829 1  |-  ( A  e.  B  ->  { A }  e.  ~P B
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1767   _Vcvv 3118    C_ wss 3481   ~Pcpw 4016   {csn 4033
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-pw 4018  df-sn 4034  df-pr 4036  df-vd1 32828
This theorem is referenced by: (None)
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