Theorem elpw 3621

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 3215	. 2 |- ( x = A -> ( x C_ B <-> A C_ B ) )
3		df-pw 3617	. 2 |- ~P B = { x | x C_ B }
4	1, 2, 3	elab2 2920	1 |- ( A e. ~P B <-> A C_ B )

Syntax hints:   <-> wb 105   e. wcel 2175   _Vcvv 2771   C_ wss 3165   ~Pcpw 3615

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-in 3171  df-ss 3178  df-pw 3617

This theorem is referenced by:  velpw  3622  elpwg  3623  prsspw  3805  pwprss  3845  pwtpss  3846  pwv  3848  sspwuni  4011  iinpw  4017  iunpwss  4018  0elpw  4207  pwuni  4235  snelpw  4256  sspwb  4259  ssextss  4263  pwin  4328  pwunss  4329  iunpw  4526  xpsspw  4786  ssenen  6947  pw1ne3  7341  3nsssucpw1  7347  ioof  10092  tgdom  14515  distop  14528  epttop  14533  resttopon  14614  txuni2  14699