Let $M,C,P,F,$ be nonempty sets satisfying the following conditions:

1. $M\subset C$;
2. $M\cap P\neq \emptyset$;
3. $C\cap F\neq \emptyset$;
4. $F\subset C\cup P$;
5. $P\cap C^{c}\neq \emptyset$.

Is it true that $F\subset M\cup P?$ I was told by a friend of mine that it is true.

I wasn't able to solve that. If I start by saying that if $x\in F$, then by (4) I get $x\in C$ or $x\in P$ and I got stuck. Then I'tried another way. If $x\notin M\cup P$ then I get $x\notin M$ or $x\notin P$, but again, I don't know how use all the hypothesis.

It’s not necessarily true. Let $M=\{1\}$, $C=\{1,2\}$, $P=\{1,3\}$, $F=\{2\}$; all five conditions are satisfied, but $F\cap(M\cup P)=\varnothing$.