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Theorem elpw 3517
Description: Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 31-Dec-1993.)
Hypothesis
Ref Expression
elpw.1  |-  A  e. 
_V
Assertion
Ref Expression
elpw  |-  ( A  e.  ~P B  <->  A  C_  B
)

Proof of Theorem elpw
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elpw.1 . 2  |-  A  e. 
_V
2 sseq1 3121 . 2  |-  ( x  =  A  ->  (
x  C_  B  <->  A  C_  B
) )
3 df-pw 3513 . 2  |-  ~P B  =  { x  |  x 
C_  B }
41, 2, 3elab2 2833 1  |-  ( A  e.  ~P B  <->  A  C_  B
)
Colors of variables: wff set class
Syntax hints:    <-> wb 104    e. wcel 1481   _Vcvv 2687    C_ wss 3072   ~Pcpw 3511
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-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
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 2689  df-in 3078  df-ss 3085  df-pw 3513
This theorem is referenced by:  velpw  3518  elpwg  3519  prsspw  3696  pwprss  3736  pwtpss  3737  pwv  3739  sspwuni  3901  iinpw  3907  iunpwss  3908  0elpw  4092  pwuni  4120  snelpw  4139  sspwb  4142  ssextss  4146  pwin  4208  pwunss  4209  iunpw  4405  xpsspw  4655  ssenen  6749  ioof  9780  tgdom  12271  distop  12284  epttop  12289  resttopon  12370  txuni2  12455
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