Theorem 3 - If $A\subset B$ and $B\subset C$, then $A\subset C$
Proof: By Remark 1, we may assume $A$ is non-empty. Let $x\in A$.
Now, since $A\subset B$, then $x\in B$. And since $B\subset C$, $x\in C$.
Since $x$ is arbitrary, if $x\in A$, then $x\in C$ implies that $A\subset C$.
