Theorem elpw2 4247

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 4246	. 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 2202   _Vcvv 2802   C_ wss 3200   ~Pcpw 3652

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213  ax-sep 4207

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-in 3206  df-ss 3213  df-pw 3654

This theorem is referenced by:  elpwi2  4248  axpweq  4261  genpelxp  7731  ltexprlempr  7828  recexprlempr  7852  cauappcvgprlemcl  7873  cauappcvgprlemladd  7878  caucvgprlemcl  7896  caucvgprprlemcl  7924  uzf  9758  ixxf  10133  fzf  10247  cncfval  15299  reldvg  15406  dvfvalap  15408  plyval  15459