Theorem pwexb 4390

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 4389	. 2 |- ( ~P A e. _V <-> U. ~P A e. _V )
2		unipw 4134	. . 3 |- U. ~P A = A
3	2	eleq1i 2203	. 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 1480   _Vcvv 2681   ~Pcpw 3505   U.cuni 3731

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-un 4350

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420  df-v 2683  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-uni 3732

This theorem is referenced by: (None)