Let $X$ be a D11: Set.
A D11: Set $\mathcal{S} \subseteq \mathcal{P}(X)$ is a proper set partition of $X$ if and only if

(1)	\begin{equation} X = \cup \mathcal{S} \end{equation}	D4984: Tight set cover
(2)	\begin{equation} \forall \, E, F \in \mathcal{S} \left( E \neq F \quad \implies \quad E \cap F = \emptyset \right) \end{equation}	D1681: Disjoint set collection
(3)	\begin{equation} \forall \, E \in \mathcal{S} : E \neq \emptyset \end{equation}

Let $X$ be a D11: Set.
A D5143: Set partition $\mathcal{S} \subseteq \mathcal{P}(X)$ of $X$ is a proper set partition of $X$ if and only if \begin{equation} \forall \, E \in \mathcal{S} : E \neq \emptyset \end{equation}