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Theorem bj-pwvrelb 36881
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 3499 . 2 (𝐴 ∈ 𝒫 (V × V) → 𝐴 ∈ V)
2 pwvrel 5739 . 2 (𝐴 ∈ V → (𝐴 ∈ 𝒫 (V × V) ↔ Rel 𝐴))
31, 2biadanii 822 1 (𝐴 ∈ 𝒫 (V × V) ↔ (𝐴 ∈ V ∧ Rel 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wcel 2106  Vcvv 3478  𝒫 cpw 4605   × cxp 5687  Rel wrel 5694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-ss 3980  df-pw 4607  df-rel 5696
This theorem is referenced by: (None)
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