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Theorem elpw 3431
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 3045 . 2  |-  ( x  =  A  ->  (
x  C_  B  <->  A  C_  B
) )
3 df-pw 3427 . 2  |-  ~P B  =  { x  |  x 
C_  B }
41, 2, 3elab2 2761 1  |-  ( A  e.  ~P B  <->  A  C_  B
)
Colors of variables: wff set class
Syntax hints:    <-> wb 103    e. wcel 1438   _Vcvv 2619    C_ wss 2997   ~Pcpw 3425
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-in 3003  df-ss 3010  df-pw 3427
This theorem is referenced by:  selpw  3432  elpwg  3433  prsspw  3604  pwprss  3644  pwtpss  3645  pwv  3647  sspwuni  3808  iinpw  3811  iunpwss  3812  0elpw  3991  pwuni  4018  snelpw  4031  sspwb  4034  ssextss  4038  pwin  4100  pwunss  4101  iunpw  4292  xpsspw  4538  ssenen  6547  ioof  9358
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