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Theorem elpw2g 4130
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 3563 . 2  |-  ( A  e.  ~P B  ->  A  C_  B )
2 ssexg 4116 . . . 4  |-  ( ( A  C_  B  /\  B  e.  V )  ->  A  e.  _V )
3 elpwg 3562 . . . . 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 2135   _Vcvv 2722    C_ wss 3112   ~Pcpw 3554
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 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146  ax-sep 4095
This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2724  df-in 3118  df-ss 3125  df-pw 3556
This theorem is referenced by:  elpw2  4131  pwnss  4133  ifelpwung  4454  elfir  6930  istopg  12564  uniopn  12566  iscld  12670  ntrval  12677  clsval  12678  discld  12703  neival  12710  isnei  12711  restdis  12751  cnpfval  12762  cndis  12808  blfvalps  12952  blfps  12976  blf  12977  reldvg  13215
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