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.