Ordering is Equivalent to Subset Relation

Theorem
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.