Theorem elpwg 3658

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 2292	. 2 |- ( x = A -> ( x e. ~P B <-> A e. ~P B ) )
2		sseq1 3248	. 2 |- ( x = A -> ( x C_ B <-> A C_ B ) )
3		vex 2803	. . 3 |- x e. _V
4	3	elpw 3656	. 2 |- ( x e. ~P B <-> x C_ B )
5	1, 2, 4	vtoclbg 2863	1 |- ( A e. V -> ( A e. ~P B <-> A C_ B ) )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2200   C_ wss 3198   ~Pcpw 3650

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2802  df-in 3204  df-ss 3211  df-pw 3652

This theorem is referenced by:  elpwi  3659  elpwb  3660  pwidg  3664  prsspwg  3831  elpw2g  4244  snelpwg  4300  snelpwi  4301  prelpw  4303  prelpwi  4304  pwel  4308  eldifpw  4572  f1opw2  6224  2pwuninelg  6444  tfrlemibfn  6489  tfr1onlembfn  6505  tfrcllembfn  6518  elpmg  6828  pw2f1odclem  7015  fopwdom  7017  fiinopn  14718  ssntr  14836  incistruhgr  15931  upgr1edc  15962  uspgr1edc  16079