[Math] For any sets $A, B$, and $C$, if $A \subseteq B$ and $A \subseteq C$, then $A \subseteq (B \cap C)$

Question

I made a Venn Diagram so I know that this is true. Now I just need some help on getting the proof right.

What I did first was obviously assume the premises, and then I tried to unpack them. So now I have $x\in A \to x\in B$ because $A \subseteq B$ and $x\in A \to x\in C$ because $A \subseteq C$.

Any pointers on where I should go from here?

Answer 1

Let $x$ be an arbitrary element of $A$. We want to conclude that $x$ also belongs to $B \cap C$. (This will show $A \subseteq B \cap C$.)

Now, since $A \subseteq B$, we know $x \in B$. Similarly, $x \in C$. Since $x \in B$ and $x \in C$, this means $x \in B \cap C$, as desired. (The set $B \cap C$ is defined to be the collection of all elements belonging to both $B$ and $C$.)