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Theorem snelpwrVD 28922
Description: Virtual deduction proof of snelpwi 4236. (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 4232 . . 3  |-  { A }  e.  _V
2 idn1 28641 . . . 4  |-  (. A  e.  B  ->.  A  e.  B ).
3 snssi 3775 . . . 4  |-  ( A  e.  B  ->  { A }  C_  B )
42, 3e1_ 28704 . . 3  |-  (. A  e.  B  ->.  { A }  C_  B ).
5 elpwg 3645 . . . 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 28768 . 2  |-  (. A  e.  B  ->.  { A }  e.  ~P B ).
87in1 28638 1  |-  ( A  e.  B  ->  { A }  e.  ~P B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1696   _Vcvv 2801    C_ wss 3165   ~Pcpw 3638   {csn 3653
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-pw 3640  df-sn 3659  df-pr 3660  df-vd1 28637
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