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Theorem elpw2 3993
Description: Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 11-Oct-2007.)
Hypothesis
Ref Expression
elpw2.1  |-  B  e. 
_V
Assertion
Ref Expression
elpw2  |-  ( A  e.  ~P B  <->  A  C_  B
)

Proof of Theorem elpw2
StepHypRef Expression
1 elpw2.1 . 2  |-  B  e. 
_V
2 elpw2g 3992 . 2  |-  ( B  e.  _V  ->  ( A  e.  ~P B  <->  A 
C_  B ) )
31, 2ax-mp 7 1  |-  ( A  e.  ~P B  <->  A  C_  B
)
Colors of variables: wff set class
Syntax hints:    <-> 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:  axpweq  4006  genpelxp  7070  ltexprlempr  7167  recexprlempr  7191  cauappcvgprlemcl  7212  cauappcvgprlemladd  7217  caucvgprlemcl  7235  caucvgprprlemcl  7263  uzf  9022  ixxf  9316  fzf  9428  cncfval  11628
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