Theorem pwvabrel 5706

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 5705	. . 3 ⊢ (𝑥 ∈ V → (𝑥 ∈ 𝒫 (V × V) ↔ Rel 𝑥))
2	1	elv 3459	. 2 ⊢ (𝑥 ∈ 𝒫 (V × V) ↔ Rel 𝑥)
3	2	eqabi 2899	1 ⊢ 𝒫 (V × V) = {𝑥 ∣ Rel 𝑥}

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570   ∈ wcel 2147  {cab 2741  Vcvv 3454  𝒫 cpw 4560   × cxp 5653  Rel wrel 5660

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-ss 3901  df-pw 4562  df-rel 5662

This theorem is used by: (None)