Theorem pwexb 2915

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 <-> P~A e. V)

Proof of Theorem pwexb

Step	Hyp	Ref	Expression
1		uniexb 2914	. 2 |- (P~A e. V <-> U.P~A e. V)
2		unipw 2763	. . 3 |- U.P~A = A
3	2	eleq1i 1540	. 2 |- (U.P~A e. V <-> A e. V)
4	1, 3	bitr2 174	1 |- (A e. V <-> P~A e. V)

Syntax hints:   <-> wb 146   e. wcel 960  Vcvv 1814  P~cpw 2406  U.cuni 2508

This theorem is referenced by:  pwuninel 4493  2pwuninel 4494  pwfi 4586  pwfiOLD 4587  ranklim 4702  r1pw 4703

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2709  ax-pow 2749  ax-un 2873

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2053  df-un 2054  df-in 2055  df-ss 2057  df-nul 2285  df-pw 2407  df-sn 2417  df-pr 2418  df-uni 2509

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