Prove or disprove the logic statement about subsets

Prove or disprove:

If $A \nsubseteq B \cap C$, then $A \nsubseteq B$ and $A \nsubseteq C$.

I think it is false statement, shown by this counterexample:

let $A=\{1,2,3\} , B=\{1,2,3\}$, and $C=\{4,5\}$,

so $B \cap C=\emptyset$, but $A \subseteq B$.

Is my counterexample correct?

• Yes, your counterexample is correct. – lulu Mar 10 '17 at 2:54
• Yes, it works as a counterexample – Bram28 Mar 10 '17 at 2:55
• Yes, but you don't need to mention that $A\not\subseteq C$. What you need is to mention that $A\not\subseteq B\cap C$ but $A\subseteq B$. – Juniven Mar 10 '17 at 3:01