Theorem elpw 3612

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 3207	. 2 |- ( x = A -> ( x C_ B <-> A C_ B ) )
3		df-pw 3608	. 2 |- ~P B = { x | x C_ B }
4	1, 2, 3	elab2 2912	1 |- ( A e. ~P B <-> A C_ B )

Syntax hints:   <-> wb 105   e. wcel 2167   _Vcvv 2763   C_ wss 3157   ~Pcpw 3606

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-in 3163  df-ss 3170  df-pw 3608

This theorem is referenced by:  velpw  3613  elpwg  3614  prsspw  3796  pwprss  3836  pwtpss  3837  pwv  3839  sspwuni  4002  iinpw  4008  iunpwss  4009  0elpw  4198  pwuni  4226  snelpw  4247  sspwb  4250  ssextss  4254  pwin  4318  pwunss  4319  iunpw  4516  xpsspw  4776  ssenen  6921  pw1ne3  7313  3nsssucpw1  7319  ioof  10063  tgdom  14392  distop  14405  epttop  14410  resttopon  14491  txuni2  14576