Theorem elpwg 3567

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 2229	. 2 |- ( x = A -> ( x e. ~P B <-> A e. ~P B ) )
2		sseq1 3165	. 2 |- ( x = A -> ( x C_ B <-> A C_ B ) )
3		vex 2729	. . 3 |- x e. _V
4	3	elpw 3565	. 2 |- ( x e. ~P B <-> x C_ B )
5	1, 2, 4	vtoclbg 2787	1 |- ( A e. V -> ( A e. ~P B <-> A C_ B ) )

Syntax hints:   -> wi 4   <-> wb 104   e. wcel 2136   C_ wss 3116   ~Pcpw 3559

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-in 3122  df-ss 3129  df-pw 3561

This theorem is referenced by:  elpwi  3568  elpwb  3569  pwidg  3573  prsspwg  3732  elpw2g  4135  snelpwi  4190  prelpwi  4192  pwel  4196  eldifpw  4455  f1opw2  6044  2pwuninelg  6251  tfrlemibfn  6296  tfr1onlembfn  6312  tfrcllembfn  6325  elpmg  6630  fopwdom  6802  fiinopn  12642  ssntr  12762