Theorem pwexb 1963

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 ∈ V ↔ ℘A ∈ V)

Proof of Theorem pwexb

Step	Hyp	Ref	Expression
1		uniexb 1962	. 2 ⊢ (℘A ∈ V ↔ ∪℘A ∈ V)
2		unipw 1960	. . 3 ⊢ ∪℘A = A
3	2	eleq1i 1152	. 2 ⊢ (∪℘A ∈ V ↔ A ∈ V)
4	1, 3	bitr2 152	1 ⊢ (A ∈ V ↔ ℘A ∈ V)

Syntax hints:   ↔ wb 127   ∈ wcel 1092  Vcvv 1348  ℘cpw 1798  ∪cuni 1919

This theorem is referenced by:  r1pw 3529

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-uni 1920

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