Theorem elpw 3655

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 3247	. 2 |- ( x = A -> ( x C_ B <-> A C_ B ) )
3		df-pw 3651	. 2 |- ~P B = { x | x C_ B }
4	1, 2, 3	elab2 2951	1 |- ( A e. ~P B <-> A C_ B )

Syntax hints:   <-> wb 105   e. wcel 2200   _Vcvv 2799   C_ wss 3197   ~Pcpw 3649

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-in 3203  df-ss 3210  df-pw 3651

This theorem is referenced by:  velpw  3656  elpwg  3657  prsspw  3842  pwprss  3883  pwtpss  3884  pwv  3886  sspwuni  4049  iinpw  4055  iunpwss  4056  0elpw  4247  pwuni  4275  snelpw  4297  sspwb  4301  ssextss  4305  pwin  4372  pwunss  4373  iunpw  4570  xpsspw  4830  ssenen  7008  pw1ne3  7411  3nsssucpw1  7417  ioof  10163  tgdom  14740  distop  14753  epttop  14758  resttopon  14839  txuni2  14924  umgrbien  15904  umgredg  15937