Theorem bj-pwvrelb 34241

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 3511	. 2 ⊢ (𝐴 ∈ 𝒫 (V × V) → 𝐴 ∈ V)
2		pwvrel 5599	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 (V × V) ↔ Rel 𝐴))
3	1, 2	biadanii 820	1 ⊢ (𝐴 ∈ 𝒫 (V × V) ↔ (𝐴 ∈ V ∧ Rel 𝐴))

Syntax hints:   ↔ wb 208   ∧ wa 398   ∈ wcel 2113  Vcvv 3493  𝒫 cpw 4536   × cxp 5550  Rel wrel 5557

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-v 3495  df-in 3940  df-ss 3949  df-pw 4538  df-rel 5559

This theorem is referenced by: (None)