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Theorem snelpw 4679
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 4135 . 2  |-  ( A  e.  B  <->  { A }  C_  B )
3 snex 4674 . . 3  |-  { A }  e.  _V
43elpw 3999 . 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 1802   _Vcvv 3093    C_ wss 3458   ~Pcpw 3993   {csn 4010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pr 4672
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-v 3095  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-pw 3995  df-sn 4011  df-pr 4013
This theorem is referenced by:  dis2ndc  19827  dislly  19864
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