Let $A, B, C$ be sets. Prove that if $B \subseteq C$, then $(A\cap B)\subseteq (A\cap C)$

I tried this question so I'll tell you guys what I have. Please correct me if I am wrong.

Assume $B \subseteq C$, then if $x$ is an element of $B$, $x$ must be an element of $C$.

Then if $A$ was added to set $B$ and $A$ was added to set $C$, every element in $A\cap B$ must belong to $A\cap C$.

I'm not really sure how to write that mathematically or symbolically since I'm new to this

## Best Answer

We have $A \cap B \subset A$ and $A \cap B \subset B \subset C$. Consequently, if $x \in A \cap B$, we have $x \in A$ and $x\in C$, from which we have $x \in A \cap C$. Hence $A \cap B \subset A \cap C$.