Theorem elpwg 3629

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 2269	. 2 |- ( x = A -> ( x e. ~P B <-> A e. ~P B ) )
2		sseq1 3220	. 2 |- ( x = A -> ( x C_ B <-> A C_ B ) )
3		vex 2776	. . 3 |- x e. _V
4	3	elpw 3627	. 2 |- ( x e. ~P B <-> x C_ B )
5	1, 2, 4	vtoclbg 2836	1 |- ( A e. V -> ( A e. ~P B <-> A C_ B ) )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2177   C_ wss 3170   ~Pcpw 3621

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-in 3176  df-ss 3183  df-pw 3623

This theorem is referenced by:  elpwi  3630  elpwb  3631  pwidg  3635  prsspwg  3799  elpw2g  4208  snelpwi  4264  prelpwi  4266  pwel  4270  eldifpw  4532  f1opw2  6165  2pwuninelg  6382  tfrlemibfn  6427  tfr1onlembfn  6443  tfrcllembfn  6456  elpmg  6764  pw2f1odclem  6946  fopwdom  6948  fiinopn  14551  ssntr  14669  incistruhgr  15761  upgr1edc  15789