Theorem elpw 3675

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 3261	. 2 |- ( x = A -> ( x C_ B <-> A C_ B ) )
3		df-pw 3671	. 2 |- ~P B = { x | x C_ B }
4	1, 2, 3	elab2 2965	1 |- ( A e. ~P B <-> A C_ B )

Syntax hints:   <-> wb 105   e. wcel 2203   _Vcvv 2813   C_ wss 3211   ~Pcpw 3669

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-in 3217  df-ss 3224  df-pw 3671

This theorem is referenced by:  velpw  3676  elpwg  3677  prsspw  3869  pwprss  3910  pwtpss  3911  pwv  3913  sspwuni  4076  iinpw  4082  iunpwss  4083  0elpw  4277  pwuni  4305  snelpw  4328  sspwb  4332  ssextss  4336  pwin  4403  pwunss  4404  iunpw  4601  xpsspw  4862  ssenen  7105  pw1ne3  7540  3nsssucpw1  7546  ioof  10304  hashfibclem  11206  tgdom  14937  distop  14950  epttop  14955  resttopon  15036  txuni2  15121  umgrbien  16105  umgredg  16140