Prove that if $A \subseteq B$ and $B \subset C$, then $A \subset C$. Prove that if $A \subseteq B$ and $B \subset C$, then $A \subset C$.

Proof:
$A \subseteq B \Longrightarrow \forall x\in A, x \in B.$ Since $B \subset C$, it follows that $x \in B \Longrightarrow x \in C$ but $\exists c \in C \ni c \notin B.$
Since $A \subseteq B$, it follows that $c \notin A$, thus $A \subset C$.

Is this good enough?

If A is a subset B then for all A is element of B. It follows B is a subset of C for all B is element of C. Since A is element of B for all A and B is element of C for all B Therefore A is subset of C.
