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Theorem snelpw 4699
Description: A singleton of a set belongs to the power class of a class containing the set. (Contributed by NM, 1-Apr-1998.)
Hypothesis
Ref Expression
snelpw.1  |-  A  e. 
_V
Assertion
Ref Expression
snelpw  |-  ( A  e.  B  <->  { A }  e.  ~P B
)

Proof of Theorem snelpw
StepHypRef Expression
1 snelpw.1 . . 3  |-  A  e. 
_V
21snss 4157 . 2  |-  ( A  e.  B  <->  { A }  C_  B )
3 snex 4694 . . 3  |-  { A }  e.  _V
43elpw 4022 . 2  |-  ( { A }  e.  ~P B 
<->  { A }  C_  B )
52, 4bitr4i 252 1  |-  ( A  e.  B  <->  { A }  e.  ~P B
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    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
This theorem is referenced by:  dis2ndc  19829  dislly  19866  dissnlocfin  19898
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