Theorem pwexb 4224

Description: The Axiom of Power Sets and its converse. A class is a set iff its power class is a set. (Contributed by NM, 11-Nov-2003.)

Assertion

Ref	Expression
pwexb	|- ( A e. _V <-> ~P A e. _V )

Proof of Theorem pwexb

Step	Hyp	Ref	Expression
1		uniexb 4223	. 2 |- ( ~P A e. _V <-> U. ~P A e. _V )
2		unipw 3972	. . 3 |- U. ~P A = A
3	2	eleq1i 2144	. 2 |- ( U. ~P A e. _V <-> A e. _V )
4	1, 3	bitr2i 183	1 |- ( A e. _V <-> ~P A e. _V )

Syntax hints:   <-> wb 103   e. wcel 1433   _Vcvv 2601   ~Pcpw 3382   U.cuni 3601

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-v 2603  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-uni 3602

This theorem is referenced by: (None)