TITLE: Is my proof logically sound?
QUESTION [5 upvotes]: Prove that for arbitrary sets A, B, and C that
$$ A \subseteq B \implies (A \cap C) \subseteq (A \cap B). $$
My Proof
Assume $A \subseteq B$. From this assumption we get  $x \in A \implies x \in B$. If we assume that $x \in (A \cap C)$ then $x \in A$ and $x \in C$. From our assumption, it follows that  $x \in B$. Since $x \in A$ and $x \in B$ we can conclude that $A \subseteq B \implies (A \cap C) \subseteq (A \cap B)$.

REPLY [5 votes]: This proof is correct! It can be simplified by noting that $A\cap B=A$ which directly implies the result since $A\cap C\subseteq A$. Written out completely, such an argument would go as follows.

Let $A\subseteq B$. Then $A\cap B=A$. Since $A\cap C\subseteq A$ always holds,  we get $A\cap C\subseteq A=A\cap B$.