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Theorem elpw2g 3992
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 3438 . 2  |-  ( A  e.  ~P B  ->  A  C_  B )
2 ssexg 3978 . . . 4  |-  ( ( A  C_  B  /\  B  e.  V )  ->  A  e.  _V )
3 elpwg 3437 . . . . 5  |-  ( A  e.  _V  ->  ( A  e.  ~P B  <->  A 
C_  B ) )
43biimparc 293 . . . 4  |-  ( ( A  C_  B  /\  A  e.  _V )  ->  A  e.  ~P B
)
52, 4syldan 276 . . 3  |-  ( ( A  C_  B  /\  B  e.  V )  ->  A  e.  ~P B
)
65expcom 114 . 2  |-  ( B  e.  V  ->  ( A  C_  B  ->  A  e.  ~P B ) )
71, 6impbid2 141 1  |-  ( B  e.  V  ->  ( A  e.  ~P B  <->  A 
C_  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    e. wcel 1438   _Vcvv 2619    C_ wss 2999   ~Pcpw 3429
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  ax-sep 3957
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 3005  df-ss 3012  df-pw 3431
This theorem is referenced by:  elpw2  3993  pwnss  3994  istopg  11596  uniopn  11598
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