Comments on my proof of the transitive property of subsets? Because I am in an advanced Calc III course that is quite proof-based, my university does not require that I take the "introduction to formal proofs" course before proceeding to higher level math classes because they assume the class already knows everything.  But as a personal sanity check, I'd like to attempt to prove something simple (the transitive property of subsets) and have you critique my proof, letting me know if I messed up in any way:
Theorem: Let $A$, $B$, $C$ be sets. If $A \subseteq B$ and $B \subseteq C$, then $A \subseteq C$.
Proof: Since $A \subseteq B$, we know that every element in $A$ is contained in $B$.  We also know that because $B \subseteq C$, every element in $B$, which includes every element in $A$, is contained in $C$.  Since every element in $A$ is contained in $C$, then by definition $A \subseteq C$. $\qquad\qquad\qquad\qquad\qquad \Box$

When someone says write a formal proof, what they usually mean is you do things strictly by the definitions.  By definition, $X \subseteq Y$ if and only if for any $x \in X$, we have $x \in Y$.  
Let $x \in A$.  Since $A \subseteq B$, we have $x \in B$.  Since $B \subseteq C$, we have $x \in C$.  We showed that for any $x \in A$, we have $x \in C$, so $A \subseteq C$.
