Theorem bj-pwvrelb 37143

Description: Characterization of the elements of the powerclass of the cartesian square of the universal class: they are exactly the sets which are binary relations. (Contributed by BJ, 16-Dec-2023.)

Assertion

Ref	Expression
bj-pwvrelb	⊢ (𝐴 ∈ 𝒫 (V × V) ↔ (𝐴 ∈ V ∧ Rel 𝐴))

Proof of Theorem bj-pwvrelb

Step	Hyp	Ref	Expression
1		elex 3463	. 2 ⊢ (𝐴 ∈ 𝒫 (V × V) → 𝐴 ∈ V)
2		pwvrel 5682	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 (V × V) ↔ Rel 𝐴))
3	1, 2	biadanii 822	1 ⊢ (𝐴 ∈ 𝒫 (V × V) ↔ (𝐴 ∈ V ∧ Rel 𝐴))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2114  Vcvv 3442  𝒫 cpw 4556   × cxp 5630  Rel wrel 5637

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-ss 3920  df-pw 4558  df-rel 5639

This theorem is referenced by: (None)