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  4294  snelpwg  4352  snelpwi  4353  prelpw  4355  prelpwi  4356  pwel  4360  eldifpw  4625  f1opw2  6299  2pwuninelg  6558  tfrlemibfn  6603  tfr1onlembfn  6619  tfrcllembfn  6632  elpmg  6942  pw2f1odclem  7138  fopwdom  7140  elfpw  7266  fiinopn  15230  ssntr  15348  incistruhgr  16536  upgr1edc  16567  uspgr1edc  16686  uhgrspansubgrlem  16722  eupth2lemsfi  16924