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Theorem elpwg 3398
Description: Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 6-Aug-2000.)
Assertion
Ref Expression
elpwg  |-  ( A  e.  V  ->  ( A  e.  ~P B  <->  A 
C_  B ) )

Proof of Theorem elpwg
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eleq1 2142 . 2  |-  ( x  =  A  ->  (
x  e.  ~P B  <->  A  e.  ~P B ) )
2 sseq1 3021 . 2  |-  ( x  =  A  ->  (
x  C_  B  <->  A  C_  B
) )
3 vex 2605 . . 3  |-  x  e. 
_V
43elpw 3396 . 2  |-  ( x  e.  ~P B  <->  x  C_  B
)
51, 2, 4vtoclbg 2660 1  |-  ( A  e.  V  ->  ( A  e.  ~P B  <->  A 
C_  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    e. wcel 1434    C_ wss 2974   ~Pcpw 3390
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-in 2980  df-ss 2987  df-pw 3392
This theorem is referenced by:  elpwi  3399  pwidg  3403  prsspwg  3552  elpw2g  3939  snelpwi  3975  prelpwi  3977  pwel  3981  eldifpw  4234  f1opw2  5737  2pwuninelg  5932  tfrlemibfn  5977  tfr1onlembfn  5993  tfrcllembfn  6006  fopwdom  6380
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