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Theorem pwidb 4585
Description: A class is an element of its powerclass if and only if it is a set. (Contributed by BJ, 31-Dec-2023.)
Assertion
Ref Expression
pwidb (𝐴 ∈ V ↔ 𝐴 ∈ 𝒫 𝐴)

Proof of Theorem pwidb
StepHypRef Expression
1 pwidg 4584 . 2 (𝐴 ∈ V → 𝐴 ∈ 𝒫 𝐴)
2 elex 3465 . 2 (𝐴 ∈ 𝒫 𝐴𝐴 ∈ V)
31, 2impbii 208 1 (𝐴 ∈ V ↔ 𝐴 ∈ 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wcel 2107  Vcvv 3447  𝒫 cpw 4564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3449  df-in 3921  df-ss 3931  df-pw 4566
This theorem is referenced by: (None)
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