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Theorem snelpw 4142
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 3656 . 2  |-  ( A  e.  B  <->  { A }  C_  B )
31snex 4116 . . 3  |-  { A }  e.  _V
43elpw 3520 . 2  |-  ( { A }  e.  ~P B 
<->  { A }  C_  B )
52, 4bitr4i 186 1  |-  ( A  e.  B  <->  { A }  e.  ~P B
)
Colors of variables: wff set class
Syntax hints:    <-> wb 104    e. wcel 1481   _Vcvv 2689    C_ wss 3075   ~Pcpw 3514   {csn 3531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537
This theorem is referenced by: (None)
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