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.

$\Box$

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.

$\Box$

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.

$\Box$

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

$\blacksquare$

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text I$ -- Superinduction and Well Ordering: $\S 1$ Introduction to well ordering