Does $A\cap B =\varnothing \Rightarrow B\subseteq \overline{A}$? How to prove $A\cap  B =\varnothing \Rightarrow B\subseteq \overline{A}$?
If I going by definitions, there is no $x$ s.t  $x\in A$ and $x\in B$.
But, what do we can tell about $\overline{A}$? What i'm missing? 

Suppose $B\nsubseteq \overline{A}$.
Therefore exists an element $x$ so that $x\in B$ and $x \notin \overline{A}$
Then $x\in A $ hence $x\in {A}\cap B $, that's a contradiction.
