Ordering is Equivalent to Subset Relation

Theorem
Let $\left({S, \preceq}\right)$ be an ordered set.

Then there exists a set $\mathbb S$ of subsets of $S$ such that:
 * $\left({S, \preceq}\right) \cong \left({\mathbb S, \subseteq}\right)$

where:
 * $\left({\mathbb S, \subseteq}\right)$ is the relational structure consisting of $\mathbb S$ and the subset relation
 * $\cong$ denotes order isomorphism.

Specifically:

Let
 * $\mathbb S := \left\{{{\bar\downarrow}a: a \in S}\right\}$

where ${\bar\downarrow} a$ is the lower closure of $a$. That is:
 * ${\bar\downarrow}a := \left\{{b \in S: b \preceq a}\right\}$

Let the mapping $\phi: S \to \mathbb S$ be defined as:
 * $\phi \left({a}\right) = {\bar\downarrow}a$

Then $\phi$ is an order isomorphism from $\left({S, \preceq}\right)$ to $\left({\mathbb S, \subseteq}\right)$.

Proof
From Subset Relation is Ordering, we have that $\left({\mathbb S, \subseteq}\right)$ is an ordered set.

We are to show that $\phi$ is an order isomorphism.

$\phi$ is clearly surjective, as every ${\bar\downarrow}a$ is defined from some $a \in S$.

By the lemma, $\phi$ is order-preserving.

Suppose that ${\bar\downarrow}a_1 \subseteq {\bar\downarrow}a_2$.

Then, since $a_1 \in {\bar\downarrow}a_1$, also $a_1 \in {\bar\downarrow}a_2$ by definition of subset.

By definition of ${\bar\downarrow}a_2$, this means $a_1 \preceq a_2$.

Thus $\phi$ is also order-reflecting.

Thus it follows that $\phi$ is an order isomorphism between $\left({S, \preceq}\right)$ and $\left({\mathbb S, \subseteq}\right)$.