Theorem elpwg 3574

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 2233	. 2 |- ( x = A -> ( x e. ~P B <-> A e. ~P B ) )
2		sseq1 3170	. 2 |- ( x = A -> ( x C_ B <-> A C_ B ) )
3		vex 2733	. . 3 |- x e. _V
4	3	elpw 3572	. 2 |- ( x e. ~P B <-> x C_ B )
5	1, 2, 4	vtoclbg 2791	1 |- ( A e. V -> ( A e. ~P B <-> A C_ B ) )

Syntax hints:   -> wi 4   <-> wb 104   e. wcel 2141   C_ wss 3121   ~Pcpw 3566

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134  df-pw 3568

This theorem is referenced by:  elpwi  3575  elpwb  3576  pwidg  3580  prsspwg  3739  elpw2g  4142  snelpwi  4197  prelpwi  4199  pwel  4203  eldifpw  4462  f1opw2  6055  2pwuninelg  6262  tfrlemibfn  6307  tfr1onlembfn  6323  tfrcllembfn  6336  elpmg  6642  fopwdom  6814  fiinopn  12796  ssntr  12916