Theorem elpw 3559

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 3160	. 2 |- ( x = A -> ( x C_ B <-> A C_ B ) )
3		df-pw 3555	. 2 |- ~P B = { x | x C_ B }
4	1, 2, 3	elab2 2869	1 |- ( A e. ~P B <-> A C_ B )

Syntax hints:   <-> wb 104   e. wcel 2135   _Vcvv 2721   C_ wss 3111   ~Pcpw 3553

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2723  df-in 3117  df-ss 3124  df-pw 3555

This theorem is referenced by:  velpw  3560  elpwg  3561  prsspw  3739  pwprss  3779  pwtpss  3780  pwv  3782  sspwuni  3944  iinpw  3950  iunpwss  3951  0elpw  4137  pwuni  4165  snelpw  4185  sspwb  4188  ssextss  4192  pwin  4254  pwunss  4255  iunpw  4452  xpsspw  4710  ssenen  6808  pw1ne3  7177  3nsssucpw1  7183  ioof  9898  tgdom  12613  distop  12626  epttop  12631  resttopon  12712  txuni2  12797