Theorem pwexb 4521

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 4520	. 2 |- ( ~P A e. _V <-> U. ~P A e. _V )
2		unipw 4261	. . 3 |- U. ~P A = A
3	2	eleq1i 2271	. 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 2176   _Vcvv 2772   ~Pcpw 3616   U.cuni 3850

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-un 4480

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-v 2774  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-uni 3851

This theorem is referenced by: (None)