Theorem elpw 3463

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 3070	. 2 |- ( x = A -> ( x C_ B <-> A C_ B ) )
3		df-pw 3459	. 2 |- ~P B = { x | x C_ B }
4	1, 2, 3	elab2 2785	1 |- ( A e. ~P B <-> A C_ B )

Syntax hints:   <-> wb 104   e. wcel 1448   _Vcvv 2641   C_ wss 3021   ~Pcpw 3457

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-in 3027  df-ss 3034  df-pw 3459

This theorem is referenced by:  selpw  3464  elpwg  3465  prsspw  3639  pwprss  3679  pwtpss  3680  pwv  3682  sspwuni  3843  iinpw  3849  iunpwss  3850  0elpw  4028  pwuni  4056  snelpw  4073  sspwb  4076  ssextss  4080  pwin  4142  pwunss  4143  iunpw  4339  xpsspw  4589  ssenen  6674  ioof  9595  tgdom  12023  distop  12036  epttop  12041  resttopon  12122  txuni2  12206