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Theorem elpwg 3582
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 2240 . 2  |-  ( x  =  A  ->  (
x  e.  ~P B  <->  A  e.  ~P B ) )
2 sseq1 3178 . 2  |-  ( x  =  A  ->  (
x  C_  B  <->  A  C_  B
) )
3 vex 2740 . . 3  |-  x  e. 
_V
43elpw 3580 . 2  |-  ( x  e.  ~P B  <->  x  C_  B
)
51, 2, 4vtoclbg 2798 1  |-  ( A  e.  V  ->  ( A  e.  ~P B  <->  A 
C_  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    e. wcel 2148    C_ wss 3129   ~Pcpw 3574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-in 3135  df-ss 3142  df-pw 3576
This theorem is referenced by:  elpwi  3583  elpwb  3584  pwidg  3588  prsspwg  3750  elpw2g  4153  snelpwi  4208  prelpwi  4210  pwel  4214  eldifpw  4473  f1opw2  6070  2pwuninelg  6277  tfrlemibfn  6322  tfr1onlembfn  6338  tfrcllembfn  6351  elpmg  6657  fopwdom  6829  fiinopn  13135  ssntr  13255
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