Let $X$ be a set, and let $\mathcal{A} \subset \mathcal{P}(X)$ be a family of subsets of $X$. In the partially ordered set $(\mathcal{P}(X), \subset)$, the supremum of $\mathcal{A}$ exists and is given by
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\begin{align*}
\sup \mathcal{A} = \bigcup_{B \in \mathcal{A}} B.
\end{align*}