Theorem pwexb 4428

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 4427	. 2 |- ( ~P A e. _V <-> U. ~P A e. _V )
2		unipw 4172	. . 3 |- U. ~P A = A
3	2	eleq1i 2220	. 2 |- ( U. ~P A e. _V <-> A e. _V )
4	1, 3	bitr2i 184	1 |- ( A e. _V <-> ~P A e. _V )

Syntax hints:   <-> wb 104   e. wcel 2125   _Vcvv 2709   ~Pcpw 3539   U.cuni 3768

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2127  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130  ax-un 4388

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-rex 2438  df-v 2711  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-uni 3769

This theorem is referenced by: (None)