Theorem elpwg 3584

Description: Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 6-Aug-2000.)

Assertion

Ref	Expression
elpwg	|- ( A e. V -> ( A e. ~P B <-> A C_ B ) )

Proof of Theorem elpwg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2240	. 2 |- ( x = A -> ( x e. ~P B <-> A e. ~P B ) )
2		sseq1 3179	. 2 |- ( x = A -> ( x C_ B <-> A C_ B ) )
3		vex 2741	. . 3 |- x e. _V
4	3	elpw 3582	. 2 |- ( x e. ~P B <-> x C_ B )
5	1, 2, 4	vtoclbg 2799	1 |- ( A e. V -> ( A e. ~P B <-> A C_ B ) )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2148   C_ wss 3130   ~Pcpw 3576

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-in 3136  df-ss 3143  df-pw 3578

This theorem is referenced by:  elpwi  3585  elpwb  3586  pwidg  3590  prsspwg  3753  elpw2g  4157  snelpwi  4213  prelpwi  4215  pwel  4219  eldifpw  4478  f1opw2  6077  2pwuninelg  6284  tfrlemibfn  6329  tfr1onlembfn  6345  tfrcllembfn  6358  elpmg  6664  fopwdom  6836  fiinopn  13507  ssntr  13625