Theorem elpw 3581

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 3577	. 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 3575

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 3577

This theorem is referenced by:  velpw  3582  elpwg  3583  prsspw  3765  pwprss  3805  pwtpss  3806  pwv  3808  sspwuni  3971  iinpw  3977  iunpwss  3978  0elpw  4164  pwuni  4192  snelpw  4213  sspwb  4216  ssextss  4220  pwin  4282  pwunss  4283  iunpw  4480  xpsspw  4738  ssenen  6850  pw1ne3  7228  3nsssucpw1  7234  ioof  9970  tgdom  13542  distop  13555  epttop  13560  resttopon  13641  txuni2  13726