Theorem elpw 3611

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 3206	. 2 |- ( x = A -> ( x C_ B <-> A C_ B ) )
3		df-pw 3607	. 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 3605

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 3607

This theorem is referenced by:  velpw  3612  elpwg  3613  prsspw  3795  pwprss  3835  pwtpss  3836  pwv  3838  sspwuni  4001  iinpw  4007  iunpwss  4008  0elpw  4197  pwuni  4225  snelpw  4246  sspwb  4249  ssextss  4253  pwin  4317  pwunss  4318  iunpw  4515  xpsspw  4775  ssenen  6912  pw1ne3  7297  3nsssucpw1  7303  ioof  10046  tgdom  14308  distop  14321  epttop  14326  resttopon  14407  txuni2  14492