Theorem elpw 3691

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 3271	. 2 |- ( x = A -> ( x C_ B <-> A C_ B ) )
3		df-pw 3687	. 2 |- ~P B = { x | x C_ B }
4	1, 2, 3	elab2 2974	1 |- ( A e. ~P B <-> A C_ B )

Syntax hints:   <-> wb 105   e. wcel 2209   _Vcvv 2821   C_ wss 3220   ~Pcpw 3685

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-ss 3233  df-pw 3687

This theorem is referenced by:  velpw  3692  elpwg  3693  prsspw  3885  pwprss  3926  pwtpss  3927  pwv  3929  sspwuni  4092  iinpw  4098  iunpwss  4099  0elpw  4296  pwuni  4324  snelpw  4347  sspwb  4351  ssextss  4355  pwin  4422  pwunss  4423  iunpw  4621  xpsspw  4882  ssenen  7142  pw1ne3  7579  3nsssucpw1  7585  ioof  10352  hashfibclem  11260  ballotfilemth  13259  tgdom  15096  distop  15109  epttop  15114  resttopon  15195  txuni2  15280  umgrbien  16265  umgredg  16300