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Theorem elpw 3393
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 2994 . 2  |-  ( x  =  A  ->  (
x  C_  B  <->  A  C_  B
) )
3 df-pw 3389 . 2  |-  ~P B  =  { x  |  x 
C_  B }
41, 2, 3elab2 2713 1  |-  ( A  e.  ~P B  <->  A  C_  B
)
Colors of variables: wff set class
Syntax hints:    <-> wb 102    e. wcel 1409   _Vcvv 2574    C_ wss 2945   ~Pcpw 3387
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-in 2952  df-ss 2959  df-pw 3389
This theorem is referenced by:  selpw  3394  elpwg  3395  prsspw  3564  pwprss  3604  pwtpss  3605  pwv  3607  sspwuni  3767  iinpw  3770  iunpwss  3771  0elpw  3945  pwuni  3971  snelpw  3977  sspwb  3980  ssextss  3984  pwin  4047  pwunss  4048  iunpw  4239  xpsspw  4478  ioof  8941
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