Subset_Relation_on_Power_Set_is_Partial_Ordering

Let $S$ be a set. Let $\powerset S$ be the power set of $S$. Let $\struct {\powerset S, \subseteq}$ be the relational structure defined on $\powerset S$ by the relation $\subseteq$. Then $\struct {\powerset S, \subseteq}$ is an ordered set. The ordering $\subseteq$ is partial $S$ is neither empty nor a singleton; otherwise it is total.