Proving $ C \subset A$ and $D \subset B \rightarrow C\cap D\subset A \cap B$ How would I show the following?
Proposition
$C \subset A$ and $D \subset B \rightarrow C\cap D\subset A \cap B$
Attempt:
Let x any object.
$x\in C \rightarrow x \in A$ 
As A contain C
$x \in D \rightarrow x\in B$  
As B contains D.
if $x \in  C\cap D$ 
Then x is in C and x is in D.
And since A contain C. B contain D.
$x\in C\cap D$ then $x \in A \cap B$

Suppose that $x \in C \cap D$ then it follows that $x \in C \subset A$ and $x\in D \subset B$ hence that $x\in A$ and $x\in B$ therefore
$$x \in A\cap B$$ 
But this is true for any $x \in C \cap D$ hence $C \cap D \subset A \cap B$, as desired.
