Theorem pwexb 3852

Description: The Axiom of Power Sets and its converse. A class is a set iff its power class is a set.

Assertion

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

Proof of Theorem pwexb

Step	Hyp	Ref	Expression
1		uniexb 3851	. 2 |- (~PA e. _V <-> U.~PA e. _V)
2		unipw 3504	. . 3 |- U.~PA = A
3	2	eleq1i 1960	. 2 |- (U.~PA e. _V <-> A e. _V)
4	1, 3	bitr2i 191	1 |- (A e. _V <-> ~PA e. _V)

Syntax hints:   <-> wb 163   e. wcel 1300  _Vcvv 2292  ~Pcpw 3032  U.cuni 3177

This theorem is referenced by:  pwuninel 5550  2pwuninel 5551  pwfi 5661  ranklim 5796  r1pw 5797

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-dif 2597  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-uni 3178

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