Theorem elpw 3694

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 3690	. 2 |- ~P B = { x | x C_ B }
4	1, 2, 3	elab2 2974	1 |- ( A e. ~P B <-> A C_ B )

This proof depends on syntax axioms:   <-> wb 105   e. wcel 2209   _Vcvv 2821   C_ wss 3220   ~Pcpw 3688

This proof depends on 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 proof 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 3690

This theorem is used by:  velpw  3695  elpwg  3696  prsspw  3890  pwprss  3931  pwtpss  3932  pwv  3934  sspwuni  4097  iinpw  4103  iunpwss  4104  0elpw  4303  pwuni  4331  snelpw  4354  sspwb  4358  ssextss  4362  pwin  4429  pwunss  4430  iunpw  4628  xpsspw  4889  ssenen  7158  pw1ne3  7595  3nsssucpw1  7601  ioof  10389  hashfibclem  11303  ballotfilemth  13338  tgdom  15315  distop  15328  epttop  15333  resttopon  15414  txuni2  15499  umgrbien  16573  umgredg  16608