Theorem elpw 3388

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. ~P B <-> 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 3020	. 2 |- ( x = A -> ( x C_ B <-> A C_ B ) )
3		df-pw 3384	. 2 |- ~P B = { x | x C_ B }
4	1, 2, 3	elab2 2741	1 |- ( A e. ~P B <-> A C_ B )

Syntax hints:   <-> wb 103   e. wcel 1433   _Vcvv 2601   C_ wss 2973   ~Pcpw 3382

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-in 2979  df-ss 2986  df-pw 3384

This theorem is referenced by:  selpw  3389  elpwg  3390  prsspw  3557  pwprss  3597  pwtpss  3598  pwv  3600  sspwuni  3760  iinpw  3763  iunpwss  3764  0elpw  3938  pwuni  3963  snelpw  3968  sspwb  3971  ssextss  3975  pwin  4037  pwunss  4038  iunpw  4229  xpsspw  4468  ioof  8994