What are the steps of proving this?
If $A\cap B' = \varnothing$ then $A \subseteq B$
where $B'$ is the complement of $B$.
Let me give it a try:
Note: The conditions that $A\cap B'=\varnothing$ and that $A\subseteq B$ are in fact equivalent.
Since you are asking for the steps for proving the statement, I would like to say that write down the definitions of $$ A\cap B^c=\emptyset ,$$ $$B^c,$$ and $$ A\subset B $$ first. And then try to go on. (Here $B^c$ is your $B'$.)
Nana proves it directly. A proof by contradiction is supplied here.
If $A = \emptyset$, then $A \subseteq B$.
If $A \neq \emptyset$, let $x \in A$. We want to show that $x \in B$.
Suppose not, then $x \in B'$. We have $x \in A \cap B' = \emptyset$ which is not possible.
Hence $A \subseteq B$.
Edit: I answered this question to test the proof approach and technique I learnt from school years ago. Would anybody point out anything that is wrong or inappropriate in the proof so that I can improve them? Thanks.