Theorem elpw 3582

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 3179	. 2 |- ( x = A -> ( x C_ B <-> A C_ B ) )
3		df-pw 3578	. 2 |- ~P B = { x | x C_ B }
4	1, 2, 3	elab2 2886	1 |- ( A e. ~P B <-> A C_ B )

Syntax hints:   <-> wb 105   e. wcel 2148   _Vcvv 2738   C_ wss 3130   ~Pcpw 3576

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 2740  df-in 3136  df-ss 3143  df-pw 3578

This theorem is referenced by:  velpw  3583  elpwg  3584  prsspw  3766  pwprss  3806  pwtpss  3807  pwv  3809  sspwuni  3972  iinpw  3978  iunpwss  3979  0elpw  4165  pwuni  4193  snelpw  4214  sspwb  4217  ssextss  4221  pwin  4283  pwunss  4284  iunpw  4481  xpsspw  4739  ssenen  6851  pw1ne3  7229  3nsssucpw1  7235  ioof  9971  tgdom  13575  distop  13588  epttop  13593  resttopon  13674  txuni2  13759