Theorem elpwg 3359

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. ~PB <-> A C_ B))

Proof of Theorem elpwg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2097	. 2 |- (x = A -> (x e. ~PB <-> A e. ~PB))
2		sseq1 2960	. 2 |- (x = A -> (x C_ B <-> A C_ B))
3		vex 2554	. . 3 |- x e. _V
4	3	elpw 3357	. 2 |- (x e. ~PB <-> x C_ B)
5	1, 2, 4	vtoclbg 2608	1 |- (A e. V -> (A e. ~PB <-> A C_ B))

Syntax hints:   -> wi 4   <-> wb 98   e. wcel 1390   C_ wss 2911   ~Pcpw 3351

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-ss 2925  df-pw 3353

This theorem is referenced by:  elpwi  3360  pwidg  3364  prsspwg  3514  elpw2g  3901  snelpwi  3939  prelpwi  3941  pwel  3945  eldifpw  4174  f1opw2  5648  2pwuninelg  5839  tfrlemibfn  5883  fopwdom  6246