Theorem pwidb 4574

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 4572	. 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ 𝒫 𝐴)
2		elex 3474	. 2 ⊢ (𝐴 ∈ 𝒫 𝐴 → 𝐴 ∈ V)
3	1, 2	impbii 211	1 ⊢ (𝐴 ∈ V ↔ 𝐴 ∈ 𝒫 𝐴)

Syntax hints:   ↔ wb 208   ∈ wcel 2141  Vcvv 3453  𝒫 cpw 4552

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-ss 3919  df-pw 4554

This theorem is referenced by: (None)