Theorem bj-pwvrelb 36900

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 3500	. 2 ⊢ (𝐴 ∈ 𝒫 (V × V) → 𝐴 ∈ V)
2		pwvrel 5734	. 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 2107  Vcvv 3479  𝒫 cpw 4599   × cxp 5682  Rel wrel 5689

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  df-rel 5691

This theorem is referenced by: (None)