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Theorem snelpwrVD 31401
Description: Virtual deduction proof of snelpwi 4534. (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 4530 . . 3  |-  { A }  e.  _V
2 idn1 31120 . . . 4  |-  (. A  e.  B  ->.  A  e.  B ).
3 snssi 4014 . . . 4  |-  ( A  e.  B  ->  { A }  C_  B )
42, 3e1_ 31183 . . 3  |-  (. A  e.  B  ->.  { A }  C_  B ).
5 elpwg 3865 . . . 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 31247 . 2  |-  (. A  e.  B  ->.  { A }  e.  ~P B ).
87in1 31117 1  |-  ( A  e.  B  ->  { A }  e.  ~P B
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1761   _Vcvv 2970    C_ wss 3325   ~Pcpw 3857   {csn 3874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-v 2972  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-pw 3859  df-sn 3875  df-pr 3877  df-vd1 31116
This theorem is referenced by: (None)
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