Theorem elpw2 4169

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

Step	Hyp	Ref	Expression
1		elpw2.1	. 2 |- B e. _V
2		elpw2g 4168	. 2 |- ( B e. _V -> ( A e. ~P B <-> A C_ B ) )
3	1, 2	ax-mp 5	1 |- ( A e. ~P B <-> A C_ B )

Syntax hints:   <-> wb 105   e. wcel 2158   _Vcvv 2749   C_ wss 3141   ~Pcpw 3587

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169  ax-sep 4133

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-in 3147  df-ss 3154  df-pw 3589

This theorem is referenced by:  elpwi2  4170  axpweq  4183  genpelxp  7524  ltexprlempr  7621  recexprlempr  7645  cauappcvgprlemcl  7666  cauappcvgprlemladd  7671  caucvgprlemcl  7689  caucvgprprlemcl  7717  uzf  9545  ixxf  9912  fzf  10026  cncfval  14412  reldvg  14501  dvfvalap  14503