[[Material set theory]]
# Elementhood Relation Class Axiom

The **Elementhood Relation Class Axiom** is a possible axiom of [[material set theory]] with [[Class|classes]]: #m/def/set/nbg 
$$
\begin{align*}
(\exists \chood X)(\forall u)(\forall \shood v)[(u,v) \in X \iff u \in v]
\end{align*}
$$
which is to say, there exists a [[Class]], unique by [[Axiom of Extensionality#Axiom of Extensionality for Classes|extensionality]], representing elementhood as a [[Relation set#Relation class]].[^2015]
It is necessary to fix a [[ordered pair#Construction|construction of the ordered pair]] within the theory.

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

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