Theorem pwidb 4586

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 4584	. 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ 𝒫 𝐴)
2		elex 3478	. 2 ⊢ (𝐴 ∈ 𝒫 𝐴 → 𝐴 ∈ V)
3	1, 2	impbii 212	1 ⊢ (𝐴 ∈ V ↔ 𝐴 ∈ 𝒫 𝐴)

This proof depends on syntax axioms:   ↔ wb 209   ∈ wcel 2146  Vcvv 3457  𝒫 cpw 4564

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-ss 3923  df-pw 4566

This theorem is used by: (None)