Theorem elpwg 3398

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 2142	. 2 |- ( x = A -> ( x e. ~P B <-> A e. ~P B ) )
2		sseq1 3021	. 2 |- ( x = A -> ( x C_ B <-> A C_ B ) )
3		vex 2605	. . 3 |- x e. _V
4	3	elpw 3396	. 2 |- ( x e. ~P B <-> x C_ B )
5	1, 2, 4	vtoclbg 2660	1 |- ( A e. V -> ( A e. ~P B <-> A C_ B ) )

Syntax hints:   -> wi 4   <-> wb 103   e. wcel 1434   C_ wss 2974   ~Pcpw 3390

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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-in 2980  df-ss 2987  df-pw 3392

This theorem is referenced by:  elpwi  3399  pwidg  3403  prsspwg  3552  elpw2g  3939  snelpwi  3975  prelpwi  3977  pwel  3981  eldifpw  4234  f1opw2  5737  2pwuninelg  5932  tfrlemibfn  5977  tfr1onlembfn  5993  tfrcllembfn  6006  fopwdom  6380