Subset Relation is Ordering/Class Theory

Theorem
Let $C$ be a class.

Then the subset relation $\subseteq$ is an ordering on $C$.

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
From Subset Relation is Reflexive:


 * $\forall x \in C: x \subseteq x$

So $\subseteq$ is reflexive.

Antisymmetry
From Subset Relation is Antisymmetric:
 * $\forall x, y \in C: x \subseteq y \land y \subseteq x \iff x = y$

So $\subseteq$ is antisymmetric.

Transitivity
From Subset Relation is Transitive:


 * $\forall x, y, z \in C: x \subseteq y \land y \subseteq z \implies x \subseteq z$

That is, $\subseteq$ is transitive.

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