Theorem pwvabrel 5710

Description: The powerclass of the cartesian square of the universal class is the class of all sets which are binary relations. (Contributed by BJ, 21-Dec-2023.)

Assertion

Ref	Expression
pwvabrel	⊢ 𝒫 (V × V) = {𝑥 ∣ Rel 𝑥}

Proof of Theorem pwvabrel

Step	Hyp	Ref	Expression
1		pwvrel 5709	. . 3 ⊢ (𝑥 ∈ V → (𝑥 ∈ 𝒫 (V × V) ↔ Rel 𝑥))
2	1	elv 3458	. 2 ⊢ (𝑥 ∈ 𝒫 (V × V) ↔ Rel 𝑥)
3	2	eqabi 2897	1 ⊢ 𝒫 (V × V) = {𝑥 ∣ Rel 𝑥}

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570   ∈ wcel 2145  {cab 2740  Vcvv 3453  𝒫 cpw 4560   × cxp 5657  Rel wrel 5664

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3455  df-ss 3919  df-pw 4562  df-rel 5666

This theorem is used by: (None)