Theorem elpwg 3664

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 2294	. 2 |- ( x = A -> ( x e. ~P B <-> A e. ~P B ) )
2		sseq1 3251	. 2 |- ( x = A -> ( x C_ B <-> A C_ B ) )
3		vex 2806	. . 3 |- x e. _V
4	3	elpw 3662	. 2 |- ( x e. ~P B <-> x C_ B )
5	1, 2, 4	vtoclbg 2866	1 |- ( A e. V -> ( A e. ~P B <-> A C_ B ) )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2202   C_ wss 3201   ~Pcpw 3656

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 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-in 3207  df-ss 3214  df-pw 3658

This theorem is referenced by:  elpwi  3665  elpwb  3666  pwidg  3670  prsspwg  3838  elpw2g  4251  snelpwg  4308  snelpwi  4309  prelpw  4311  prelpwi  4312  pwel  4316  eldifpw  4580  f1opw2  6239  2pwuninelg  6492  tfrlemibfn  6537  tfr1onlembfn  6553  tfrcllembfn  6566  elpmg  6876  pw2f1odclem  7063  fopwdom  7065  fiinopn  14798  ssntr  14916  incistruhgr  16014  upgr1edc  16045  uspgr1edc  16164  uhgrspansubgrlem  16200  eupth2lemsfi  16402