Theorem elpwg 3677

Description: Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 6-Aug-2000.)

Assertion

Ref	Expression
elpwg	|- ( A e. V -> ( A e. ~P B <-> A C_ B ) )

Proof of Theorem elpwg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2295	. 2 |- ( x = A -> ( x e. ~P B <-> A e. ~P B ) )
2		sseq1 3261	. 2 |- ( x = A -> ( x C_ B <-> A C_ B ) )
3		vex 2816	. . 3 |- x e. _V
4	3	elpw 3675	. 2 |- ( x e. ~P B <-> x C_ B )
5	1, 2, 4	vtoclbg 2876	1 |- ( A e. V -> ( A e. ~P B <-> A C_ B ) )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2203   C_ wss 3211   ~Pcpw 3669

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-in 3217  df-ss 3224  df-pw 3671

This theorem is referenced by:  elpwi  3678  elpwb  3679  pwidg  3686  prsspwg  3854  elpw2g  4268  snelpwg  4326  snelpwi  4327  prelpw  4329  prelpwi  4330  pwel  4334  eldifpw  4598  f1opw2  6261  2pwuninelg  6514  tfrlemibfn  6559  tfr1onlembfn  6575  tfrcllembfn  6588  elpmg  6898  pw2f1odclem  7087  fopwdom  7089  elfpw  7215  fiinopn  14869  ssntr  14987  incistruhgr  16085  upgr1edc  16116  uspgr1edc  16235  uhgrspansubgrlem  16271  eupth2lemsfi  16473