Theorem elpwg 3367

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 2100	. 2 |- ( x = A -> ( x e. ~P B <-> A e. ~P B ) )
2		sseq1 2966	. 2 |- ( x = A -> ( x C_ B <-> A C_ B ) )
3		vex 2560	. . 3 |- x e. _V
4	3	elpw 3365	. 2 |- ( x e. ~P B <-> x C_ B )
5	1, 2, 4	vtoclbg 2614	1 |- ( A e. V -> ( A e. ~P B <-> A C_ B ) )

Syntax hints:   -> wi 4   <-> wb 98   e. wcel 1393   C_ wss 2917   ~Pcpw 3359

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-pw 3361

This theorem is referenced by:  elpwi  3368  pwidg  3372  prsspwg  3523  elpw2g  3910  snelpwi  3948  prelpwi  3950  pwel  3954  eldifpw  4208  f1opw2  5706  2pwuninelg  5898  tfrlemibfn  5942  fopwdom  6310