Theorem elpwg 3623

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 2267	. 2 |- ( x = A -> ( x e. ~P B <-> A e. ~P B ) )
2		sseq1 3215	. 2 |- ( x = A -> ( x C_ B <-> A C_ B ) )
3		vex 2774	. . 3 |- x e. _V
4	3	elpw 3621	. 2 |- ( x e. ~P B <-> x C_ B )
5	1, 2, 4	vtoclbg 2833	1 |- ( A e. V -> ( A e. ~P B <-> A C_ B ) )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2175   C_ wss 3165   ~Pcpw 3615

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-in 3171  df-ss 3178  df-pw 3617

This theorem is referenced by:  elpwi  3624  elpwb  3625  pwidg  3629  prsspwg  3792  elpw2g  4199  snelpwi  4255  prelpwi  4257  pwel  4261  eldifpw  4523  f1opw2  6151  2pwuninelg  6368  tfrlemibfn  6413  tfr1onlembfn  6429  tfrcllembfn  6442  elpmg  6750  pw2f1odclem  6930  fopwdom  6932  fiinopn  14418  ssntr  14536