Theorem pwidb 4556

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

Step	Hyp	Ref	Expression
1		pwidg 4555	. 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ 𝒫 𝐴)
2		elex 3450	. 2 ⊢ (𝐴 ∈ 𝒫 𝐴 → 𝐴 ∈ V)
3	1, 2	impbii 208	1 ⊢ (𝐴 ∈ V ↔ 𝐴 ∈ 𝒫 𝐴)

Syntax hints:   ↔ wb 205   ∈ wcel 2106  Vcvv 3432  𝒫 cpw 4533

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-in 3894  df-ss 3904  df-pw 4535

This theorem is referenced by: (None)