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


*

*$M\subset C$; 

*$M\cap P\neq \emptyset$; 

*$C\cap F\neq \emptyset$;

*$F\subset C\cup P$;

*$P\cap C^{c}\neq \emptyset$.


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.
I would appreciate your help.

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