Suppose that $A, B$ and $C$ are sets such that $A\subseteq B$ and $B\subseteq C$. Show that $A\subseteq C$
My attempted try to solve the problem:
given that $A\subseteq B$, and $B\subseteq C$ then from the definition: $$ \forall x (x\in A\Rightarrow x\in B) \\ \forall x (x\in B\Rightarrow x\in C) $$ by Hypothetical Syllogism $ \forall x (x\in A\Rightarrow x\in C)$.
Hence $A\subseteq C$.
Is this proof correct?