\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}