Theorem elpwg 3696

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 3694	. 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 ) )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   e. wcel 2209   C_ wss 3220   ~Pcpw 3688

This proof depends on 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 proof 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 3690

This theorem is used by:  elpwd  3697  elpwi  3698  elpwb  3699  pwidg  3706  prsspwg  3875  elpw2g  4292  snelpwg  4350  snelpwi  4351  prelpw  4353  prelpwi  4354  pwel  4358  eldifpw  4623  f1opw2  6296  2pwuninelg  6554  tfrlemibfn  6599  tfr1onlembfn  6615  tfrcllembfn  6628  elpmg  6938  pw2f1odclem  7134  fopwdom  7136  elfpw  7262  fiinopn  15105  ssntr  15223  incistruhgr  16331  upgr1edc  16362  uspgr1edc  16481  uhgrspansubgrlem  16517  eupth2lemsfi  16719