Theorem elpw 3622

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 3216	. 2 |- ( x = A -> ( x C_ B <-> A C_ B ) )
3		df-pw 3618	. 2 |- ~P B = { x | x C_ B }
4	1, 2, 3	elab2 2921	1 |- ( A e. ~P B <-> A C_ B )

Syntax hints:   <-> wb 105   e. wcel 2176   _Vcvv 2772   C_ wss 3166   ~Pcpw 3616

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-in 3172  df-ss 3179  df-pw 3618

This theorem is referenced by:  velpw  3623  elpwg  3624  prsspw  3806  pwprss  3846  pwtpss  3847  pwv  3849  sspwuni  4012  iinpw  4018  iunpwss  4019  0elpw  4208  pwuni  4236  snelpw  4257  sspwb  4260  ssextss  4264  pwin  4329  pwunss  4330  iunpw  4527  xpsspw  4787  ssenen  6948  pw1ne3  7342  3nsssucpw1  7348  ioof  10093  tgdom  14544  distop  14557  epttop  14562  resttopon  14643  txuni2  14728