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Theorem snelpw 4214
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 3728 . 2  |-  ( A  e.  B  <->  { A }  C_  B )
31snex 4186 . . 3  |-  { A }  e.  _V
43elpw 3582 . 2  |-  ( { A }  e.  ~P B 
<->  { A }  C_  B )
52, 4bitr4i 187 1  |-  ( A  e.  B  <->  { A }  e.  ~P B
)
Colors of variables: wff set class
Syntax hints:    <-> wb 105    e. wcel 2148   _Vcvv 2738    C_ wss 3130   ~Pcpw 3576   {csn 3593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599
This theorem is referenced by: (None)
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