Theorem elpwg 3693

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 2301	. 2 |- ( x = A -> ( x e. ~P B <-> A e. ~P B ) )
2		sseq1 3271	. 2 |- ( x = A -> ( x C_ B <-> A C_ B ) )
3		vex 2824	. . 3 |- x e. _V
4	3	elpw 3691	. 2 |- ( x e. ~P B <-> x C_ B )
5	1, 2, 4	vtoclbg 2884	1 |- ( A e. V -> ( A e. ~P B <-> A C_ B ) )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2209   C_ wss 3220   ~Pcpw 3685

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 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 theorem 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 3687

This theorem is referenced by:  elpwi  3694  elpwb  3695  pwidg  3702  prsspwg  3870  elpw2g  4287  snelpwg  4345  snelpwi  4346  prelpw  4348  prelpwi  4349  pwel  4353  eldifpw  4618  f1opw2  6286  2pwuninelg  6544  tfrlemibfn  6589  tfr1onlembfn  6605  tfrcllembfn  6618  elpmg  6928  pw2f1odclem  7124  fopwdom  7126  elfpw  7252  fiinopn  15028  ssntr  15146  incistruhgr  16245  upgr1edc  16276  uspgr1edc  16395  uhgrspansubgrlem  16431  eupth2lemsfi  16633