Theorem elpw 3357

Description: Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 31-Dec-1993.)

Hypothesis

Ref	Expression
elpw.1	|- A e. _V

Assertion

Ref	Expression
elpw	|- (A e. ~PB <-> A C_ B)

Proof of Theorem elpw

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elpw.1	. 2 |- A e. _V
2		sseq1 2960	. 2 |- (x = A -> (x C_ B <-> A C_ B))
3		df-pw 3353	. 2 |- ~PB = {x | x C_ B }
4	1, 2, 3	elab2 2684	1 |- (A e. ~PB <-> A C_ B)

Syntax hints:   <-> wb 98   e. wcel 1390   _Vcvv 2551   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:  selpw  3358  elpwg  3359  prsspw  3527  pwprss  3567  pwtpss  3568  pwv  3570  sspwuni  3730  iinpw  3733  iunpwss  3734  0elpw  3908  pwuni  3934  snelpw  3940  sspwb  3943  ssextss  3947  pwin  4010  pwunss  4011  iunpw  4177  xpsspw  4393  ioof  8610