Theorem pwexg 4015

Description: Power set axiom expressed in class notation, with the sethood requirement as an antecedent. (Contributed by NM, 30-Oct-2003.)

Assertion

Ref	Expression
pwexg	|- ( A e. V -> ~P A e. _V )

Proof of Theorem pwexg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pweq 3432	. . 3 |- ( x = A -> ~P x = ~P A )
2	1	eleq1d 2156	. 2 |- ( x = A -> ( ~P x e. _V <-> ~P A e. _V ) )
3		vpwex 4014	. 2 |- ~P x e. _V
4	2, 3	vtoclg 2679	1 |- ( A e. V -> ~P A e. _V )

Syntax hints:   -> wi 4   = wceq 1289   e. wcel 1438   _Vcvv 2619   ~Pcpw 3429

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-in 3005  df-ss 3012  df-pw 3431

This theorem is referenced by:  pwexd  4016  abssexg  4017  pwex  4018  snexg  4019  pwel  4045  uniexb  4295  xpexg  4552  fabexg  5198  mapex  6409  pmvalg  6414  fopwdom  6550  ssenen  6565  toponsspwpwg  11573