Theorem elpwg 3657

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 2292	. 2 |- ( x = A -> ( x e. ~P B <-> A e. ~P B ) )
2		sseq1 3247	. 2 |- ( x = A -> ( x C_ B <-> A C_ B ) )
3		vex 2802	. . 3 |- x e. _V
4	3	elpw 3655	. 2 |- ( x e. ~P B <-> x C_ B )
5	1, 2, 4	vtoclbg 2862	1 |- ( A e. V -> ( A e. ~P B <-> A C_ B ) )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2200   C_ wss 3197   ~Pcpw 3649

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-in 3203  df-ss 3210  df-pw 3651

This theorem is referenced by:  elpwi  3658  elpwb  3659  pwidg  3663  prsspwg  3827  elpw2g  4239  snelpwg  4295  snelpwi  4296  prelpw  4298  prelpwi  4299  pwel  4303  eldifpw  4567  f1opw2  6210  2pwuninelg  6427  tfrlemibfn  6472  tfr1onlembfn  6488  tfrcllembfn  6501  elpmg  6809  pw2f1odclem  6991  fopwdom  6993  fiinopn  14672  ssntr  14790  incistruhgr  15884  upgr1edc  15915