Ordering is Equivalent to Subset Relation/Proof 1

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.

Hence any ordering on a set can be modelled uniquely by a set of subsets of that set under the subset relation.

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

For each $a \in S$, let $a^\prec$ be the lower closure of $a$.

That is:
 * $a^\prec := \left\{{b \in S: b \preceq a}\right\}$

Then let $T$ be defined as:
 * $T := \left\{{a^\prec: a \in S}\right\}$

Let the mapping $\phi: S \to T$ be defined as:
 * $\phi \left({a}\right) = a^\prec$

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

$\phi$ is clearly surjective, as every $a^\prec$ is defined from some $a \in S$.

Now suppose $x^\prec, y^\prec \in T: x^\prec = y^\prec$.

Then:
 * $\left\{{b \in S: b \preceq x}\right\} = \left\{{b \in S: b \preceq y}\right\}$

We have that:
 * $x \in x^\prec = y^\prec$ and $y \in y^\prec = x^\prec$

which means:
 * $x \preceq y$ and $y \preceq x$

So as an ordering is antisymmetric, we have $x = y$ and so $\phi$ is injective.

Hence by definition, $\phi$ is a bijection.

Now let $a_1 \preceq a_2$.

Then by definition:
 * $a_1 \in {a_2}^\prec$

Let $a_3 \in {a_1}^\prec$.

Then by definition:
 * $a_3 \preceq a_1$

As an ordering is transitive, it follows that:
 * $a_3 \preceq a_2$

and so:
 * $a_3 \in {a_2}^\prec$

So by definition of a subset:
 * ${a_1}^\prec \subseteq {a_2}^\prec$

Therefore, $\phi$ is order-preserving

Conversely, suppose that ${a_1}^\prec \subseteq {a_2}^\prec$.

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

By definition of ${a_2}^\prec$:
 * $a_1 \preceq a_2$

Hence it is seen that $\phi^{-1}$ is also order-preserving.

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