Let $P = (X, {\preceq})$ be a D1103: Partially ordered set.
Let $\mathsf{Min} = \mathsf{Min}(P)$ be the D4459: Set of internally lower-bounded sets in $P$.

Then \begin{equation} \forall \, E, F \in \mathsf{Min} \, (E \subseteq F \quad \Rightarrow \quad \min(E) \succeq \min(F)) \end{equation}