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Theorem snelpwrVD 34031
Description: Virtual deduction proof of snelpwi 4682. (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 4678 . . 3  |-  { A }  e.  _V
2 idn1 33745 . . . 4  |-  (. A  e.  B  ->.  A  e.  B ).
3 snssi 4160 . . . 4  |-  ( A  e.  B  ->  { A }  C_  B )
42, 3e1a 33807 . . 3  |-  (. A  e.  B  ->.  { A }  C_  B ).
5 elpwg 4007 . . . 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 33871 . 2  |-  (. A  e.  B  ->.  { A }  e.  ~P B ).
87in1 33742 1  |-  ( A  e.  B  ->  { A }  e.  ~P B
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1823   _Vcvv 3106    C_ wss 3461   ~Pcpw 3999   {csn 4016
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-pw 4001  df-sn 4017  df-pr 4019  df-vd1 33741
This theorem is referenced by: (None)
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