Theorem elpwg 3660

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 2294	. 2 |- ( x = A -> ( x e. ~P B <-> A e. ~P B ) )
2		sseq1 3250	. 2 |- ( x = A -> ( x C_ B <-> A C_ B ) )
3		vex 2805	. . 3 |- x e. _V
4	3	elpw 3658	. 2 |- ( x e. ~P B <-> x C_ B )
5	1, 2, 4	vtoclbg 2865	1 |- ( A e. V -> ( A e. ~P B <-> A C_ B ) )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2202   C_ wss 3200   ~Pcpw 3652

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-in 3206  df-ss 3213  df-pw 3654

This theorem is referenced by:  elpwi  3661  elpwb  3662  pwidg  3666  prsspwg  3833  elpw2g  4246  snelpwg  4302  snelpwi  4303  prelpw  4305  prelpwi  4306  pwel  4310  eldifpw  4574  f1opw2  6228  2pwuninelg  6448  tfrlemibfn  6493  tfr1onlembfn  6509  tfrcllembfn  6522  elpmg  6832  pw2f1odclem  7019  fopwdom  7021  fiinopn  14727  ssntr  14845  incistruhgr  15940  upgr1edc  15971  uspgr1edc  16090  uhgrspansubgrlem  16126