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Theorem bj-pwvrelb 34299
 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
StepHypRef Expression
1 elex 3487 . 2 (𝐴 ∈ 𝒫 (V × V) → 𝐴 ∈ V)
2 pwvrel 5579 . 2 (𝐴 ∈ V → (𝐴 ∈ 𝒫 (V × V) ↔ Rel 𝐴))
31, 2biadanii 821 1 (𝐴 ∈ 𝒫 (V × V) ↔ (𝐴 ∈ V ∧ Rel 𝐴))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   ∈ wcel 2114  Vcvv 3469  𝒫 cpw 4511   × cxp 5530  Rel wrel 5537 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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2794 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-v 3471  df-in 3915  df-ss 3925  df-pw 4513  df-rel 5539 This theorem is referenced by: (None)
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