Theorem elpw 3580

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 3178	. 2 |- ( x = A -> ( x C_ B <-> A C_ B ) )
3		df-pw 3576	. 2 |- ~P B = { x | x C_ B }
4	1, 2, 3	elab2 2885	1 |- ( A e. ~P B <-> A C_ B )

Syntax hints:   <-> wb 105   e. wcel 2148   _Vcvv 2737   C_ wss 3129   ~Pcpw 3574

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-in 3135  df-ss 3142  df-pw 3576

This theorem is referenced by:  velpw  3581  elpwg  3582  prsspw  3763  pwprss  3803  pwtpss  3804  pwv  3806  sspwuni  3968  iinpw  3974  iunpwss  3975  0elpw  4161  pwuni  4189  snelpw  4210  sspwb  4213  ssextss  4217  pwin  4279  pwunss  4280  iunpw  4477  xpsspw  4735  ssenen  6845  pw1ne3  7223  3nsssucpw1  7229  ioof  9955  tgdom  13232  distop  13245  epttop  13250  resttopon  13331  txuni2  13416