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Theorem elpw2g 4076
Description: Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 7-Aug-2000.)
Assertion
Ref Expression
elpw2g  |-  ( B  e.  V  ->  ( A  e.  ~P B  <->  A 
C_  B ) )

Proof of Theorem elpw2g
StepHypRef Expression
1 elpwi 3514 . 2  |-  ( A  e.  ~P B  ->  A  C_  B )
2 ssexg 4062 . . . 4  |-  ( ( A  C_  B  /\  B  e.  V )  ->  A  e.  _V )
3 elpwg 3513 . . . . 5  |-  ( A  e.  _V  ->  ( A  e.  ~P B  <->  A 
C_  B ) )
43biimparc 297 . . . 4  |-  ( ( A  C_  B  /\  A  e.  _V )  ->  A  e.  ~P B
)
52, 4syldan 280 . . 3  |-  ( ( A  C_  B  /\  B  e.  V )  ->  A  e.  ~P B
)
65expcom 115 . 2  |-  ( B  e.  V  ->  ( A  C_  B  ->  A  e.  ~P B ) )
71, 6impbid2 142 1  |-  ( B  e.  V  ->  ( A  e.  ~P B  <->  A 
C_  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    e. wcel 1480   _Vcvv 2681    C_ wss 3066   ~Pcpw 3505
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-in 3072  df-ss 3079  df-pw 3507
This theorem is referenced by:  elpw2  4077  pwnss  4078  elfir  6854  istopg  12155  uniopn  12157  iscld  12261  ntrval  12268  clsval  12269  discld  12294  neival  12301  isnei  12302  restdis  12342  cnpfval  12353  cndis  12399  blfvalps  12543  blfps  12567  blf  12568  reldvg  12806
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