Theorem elpw 3627

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 3220	. 2 |- ( x = A -> ( x C_ B <-> A C_ B ) )
3		df-pw 3623	. 2 |- ~P B = { x | x C_ B }
4	1, 2, 3	elab2 2925	1 |- ( A e. ~P B <-> A C_ B )

Syntax hints:   <-> wb 105   e. wcel 2177   _Vcvv 2773   C_ wss 3170   ~Pcpw 3621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-in 3176  df-ss 3183  df-pw 3623

This theorem is referenced by:  velpw  3628  elpwg  3629  prsspw  3814  pwprss  3855  pwtpss  3856  pwv  3858  sspwuni  4021  iinpw  4027  iunpwss  4028  0elpw  4219  pwuni  4247  snelpw  4269  sspwb  4273  ssextss  4277  pwin  4342  pwunss  4343  iunpw  4540  xpsspw  4800  ssenen  6968  pw1ne3  7371  3nsssucpw1  7377  ioof  10123  tgdom  14629  distop  14642  epttop  14647  resttopon  14728  txuni2  14813  umgrbien  15791  umgredg  15819