Theorem elpwg 3518

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 2202	. 2 |- ( x = A -> ( x e. ~P B <-> A e. ~P B ) )
2		sseq1 3120	. 2 |- ( x = A -> ( x C_ B <-> A C_ B ) )
3		vex 2689	. . 3 |- x e. _V
4	3	elpw 3516	. 2 |- ( x e. ~P B <-> x C_ B )
5	1, 2, 4	vtoclbg 2747	1 |- ( A e. V -> ( A e. ~P B <-> A C_ B ) )

Syntax hints:   -> wi 4   <-> wb 104   e. wcel 1480   C_ wss 3071   ~Pcpw 3510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-in 3077  df-ss 3084  df-pw 3512

This theorem is referenced by:  elpwi  3519  elpwb  3520  pwidg  3524  prsspwg  3679  elpw2g  4081  snelpwi  4134  prelpwi  4136  pwel  4140  eldifpw  4398  f1opw2  5976  2pwuninelg  6180  tfrlemibfn  6225  tfr1onlembfn  6241  tfrcllembfn  6254  elpmg  6558  fopwdom  6730  fiinopn  12171  ssntr  12291