[[Material set theory]]
# Complement Axiom for classes

The **Complement Axiom** is a possible axiom of [[Material set theory]] with [[Class|classes]][^2015]: #m/def/set/nbg 
$$
\begin{align*}
(\forall \chood X)(\exists \chood Z)(\forall  \ehood u)[u \in Z \iff u \notin X]
\end{align*}
$$
which is to say,
the complement of a class $X$ (within the universe of everything except proper classes) exists,
which by [[Axiom of Extensionality#Axiom of Extensionality for classes|extensionality]] is unique and we denote the $X^c$.
It follows immediately that we have a **universal class**
$$
\begin{align*}
V = \0^c
\end{align*}
$$

  [^2015]: 2015\. [[Sources/@mendelsonIntroductionMathematicalLogic2015|Introduction to Mathematical Logic]], §4.1, p. 236, B3

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