Theorem elpw 3656

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 3248	. 2 |- ( x = A -> ( x C_ B <-> A C_ B ) )
3		df-pw 3652	. 2 |- ~P B = { x | x C_ B }
4	1, 2, 3	elab2 2952	1 |- ( A e. ~P B <-> A C_ B )

Syntax hints:   <-> wb 105   e. wcel 2200   _Vcvv 2800   C_ wss 3198   ~Pcpw 3650

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 2802  df-in 3204  df-ss 3211  df-pw 3652

This theorem is referenced by:  velpw  3657  elpwg  3658  prsspw  3846  pwprss  3887  pwtpss  3888  pwv  3890  sspwuni  4053  iinpw  4059  iunpwss  4060  0elpw  4252  pwuni  4280  snelpw  4302  sspwb  4306  ssextss  4310  pwin  4377  pwunss  4378  iunpw  4575  xpsspw  4836  ssenen  7032  pw1ne3  7438  3nsssucpw1  7444  ioof  10196  tgdom  14786  distop  14799  epttop  14804  resttopon  14885  txuni2  14970  umgrbien  15951  umgredg  15984