Theorem pwidb 4624

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 4623	. 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ 𝒫 𝐴)
2		elex 3490	. 2 ⊢ (𝐴 ∈ 𝒫 𝐴 → 𝐴 ∈ V)
3	1, 2	impbii 208	1 ⊢ (𝐴 ∈ V ↔ 𝐴 ∈ 𝒫 𝐴)

Syntax hints:   ↔ wb 205   ∈ wcel 2099  Vcvv 3471  𝒫 cpw 4603

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3473  df-in 3954  df-ss 3964  df-pw 4605

This theorem is referenced by: (None)