Theorem pwidb 4620

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 4619	. 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ 𝒫 𝐴)
2		elex 3500	. 2 ⊢ (𝐴 ∈ 𝒫 𝐴 → 𝐴 ∈ V)
3	1, 2	impbii 209	1 ⊢ (𝐴 ∈ V ↔ 𝐴 ∈ 𝒫 𝐴)

Syntax hints:   ↔ wb 206   ∈ wcel 2107  Vcvv 3479  𝒫 cpw 4599

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-v 3481  df-ss 3967  df-pw 4601

This theorem is referenced by: (None)