Theorem bj-pwvrelb 36921

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 3485	. 2 ⊢ (𝐴 ∈ 𝒫 (V × V) → 𝐴 ∈ V)
2		pwvrel 5709	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 (V × V) ↔ Rel 𝐴))
3	1, 2	biadanii 821	1 ⊢ (𝐴 ∈ 𝒫 (V × V) ↔ (𝐴 ∈ V ∧ Rel 𝐴))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2109  Vcvv 3464  𝒫 cpw 4580   × cxp 5657  Rel wrel 5664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-v 3466  df-ss 3948  df-pw 4582  df-rel 5666

This theorem is referenced by: (None)