[[Material set theory]]
# Universal Relation Axiom

The **Universal Relation Axiom** is a possible axiom of [[Material set theory]] with [[Class|classes]][^2015]: #m/def/set/nbg 
$$
\begin{align*}
(\forall \chood X)(\exists \chood Z)(\forall u)(\forall v)[(u,v) \in Z \iff u \in X]
\end{align*}
$$
which is to say, for any class $X$ there is a class representing a [[Relation set]] between each element of $X$ and everything else in the [[Complement Axiom for classes|universal class]], 
a sort of “[[Cartesian product]]” $X \times V$.

  [^2015]: 2015\. [[Sources/@mendelsonIntroductionMathematicalLogic2015|Introduction to Mathematical Logic]], §4.1, p. 236, B5


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