Ordering_is_Equivalent_to_Subset_Relation

Let $\struct {S, \preceq}$ be an ordered set. Then there exists a set $\mathbb S$ of subsets of $S$ such that $\struct {S, \preceq} \cong \struct {\mathbb S, \subseteq}$ where $\struct {\mathbb S, \subseteq}$ is the relational structure consisting of $\mathbb S$ and the subset relation $\cong$ denotes order isomorphism. Hence any ordering on a set can be modelled uniquely by a set of subsets of that set under the subset relation.