Theorem pwexb 4539

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 4538	. 2 |- ( ~P A e. _V <-> U. ~P A e. _V )
2		unipw 4279	. . 3 |- U. ~P A = A
3	2	eleq1i 2273	. 2 |- ( U. ~P A e. _V <-> A e. _V )
4	1, 3	bitr2i 185	1 |- ( A e. _V <-> ~P A e. _V )

Syntax hints:   <-> wb 105   e. wcel 2178   _Vcvv 2776   ~Pcpw 3626   U.cuni 3864

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rex 2492  df-v 2778  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-uni 3865

This theorem is referenced by: (None)