Subset Relation is Ordering

Theorem
Let $\mathbb S$ be a set of sets.

Then $\subseteq$ is an ordering on $\mathbb S$.

In other words, let $\struct {\mathbb S, \subseteq}$ be the relational structure defined on $\mathbb S$ by the relation $\subseteq$.

Then $\struct {\mathbb S, \subseteq}$ is an ordered set.

Proof
To establish that $\subseteq$ is an ordering, we need to show that it is reflexive, antisymmetric and transitive.

So, checking in turn each of the criteria for an ordering:

Reflexivity
So $\subseteq$ is reflexive.

Antisymmetry
So $\subseteq$ is antisymmetric.

Transitivity
That is, $\subseteq$ is transitive.

So we have shown that $\subseteq$ is an ordering on $\mathbb S$.