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\begin{signature}\label{example_union}
$A\union B$ is a set.
\end{signature}
\begin{axiom}[Extensionality]\label{ext}
Suppose for all $a$ we have $a\in A$ iff $a\in B$.
Then $A = B$.
\end{axiom}
\begin{axiom}\label{union_defn}
Let $A, B$ be sets.
$a\in A\union B$ iff $a\in A$ or $a\in B$.
\end{axiom}
\begin{proposition}\label{union_comm}
$A\union B = B\union A$.
\end{proposition}
\begin{proposition}\label{union_assoc}
$(A\union B)\union C = A\union (B\union C)$.
\end{proposition}
\begin{proof}
For all $a$ we have if $a\in (A\union B)\union C$, then $a\in A\union (B\union C)$.
For all $a$ we have if $a\in A\union (B\union C)$, then $a\in (A\union B)\union C$.
\end{proof}
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