Subclass of Subclass is Subclass

Theorem

Let $A$, $B$ and $C$ be classes.

Let $A$ be a subclass of $B$.

Let $B$ be a subclass of $C$.

Then $A$ is a subclass of $C$.

Proof

Let $x \in A$ be arbitrary.

It follows by definition of subclass that $x \in B$.

It further follows by definition of subclass that $x \in C$.

So we have that $x \in A$ implies that $x \in C$.

As $x$ is arbitrary, the result follows.

$\blacksquare$