Theorem pwexb 2908

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 2907	. 2 |- (P~A e. V <-> U.P~A e. V)
2		unipw 2756	. . 3 |- U.P~A = A
3	2	eleq1i 1537	. 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 958  Vcvv 1811  P~cpw 2401  U.cuni 2503

This theorem is referenced by:  pwuninel 4486  2pwuninel 4487  pwfiOLD 4571  ranklim 4685  r1pw 4686

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-uni 2504

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