# Prove: If $A \subseteq C$ and $B \subseteq D$, then $A \cap B \subseteq C \cap D$

Is the form and correctness of my elementwise proof of this correct? I don't have any other way of getting feedback for my proofs and I want to improve.

Proof. Suppose $A, B, C, D$ are sets such that $A \subseteq C$ and $B \subseteq D$ and let $x \in A \cap B$. It has to be shown that $x \in C \cap D$.

$x \in A \cap B$ means that $x \in A$ and $x\in B$. Because $A \subseteq C$, $x \in C$ and because $B \subseteq D$, $x \in D$. Thus, $x \in C \cap D$.

Thus, if $A \subseteq C$ and $B \subseteq D$, then $A \cap B \subseteq C \cap D$.

• This is excellent. – Brian M. Scott Oct 21 '12 at 15:18
• Thanks! I just fixed an error that I made in the title. Should I elaborate more on where $x \in A \cap B$ comes from? It comes from $A \cap B \subseteq C \cap D$, correct? – Brandon Amos Oct 21 '12 at 15:22
• @BrianM.Scott Realized that too. Deleted my comment before seeing you reply. – hwhm Oct 21 '12 at 15:23
• No need to say any more: the reason for choosing $x\in A\cap B$ initially is clear just from the inclusion that you’re trying to prove. – Brian M. Scott Oct 21 '12 at 15:24
• You’re very welcome, and I agree with what Asaf wrote in the answer below. – Brian M. Scott Oct 21 '12 at 15:37