Is it true that $P(A)\leq P(A\cap B\cap C)+P(B^c)+P(C^c)$?

Question

I am reading a paper, and there is an inequality which is printed below.

It seems he uses a result similar to $P(A)\leq P(A\cap B\cap C)+P(B^c)+P(C^c)$ for measurable sets $A,B,C$. I can't conclude if this inequality holds in general.

Is it true? Can you show it?

Thanks in advance!

Answer 1

$$\begin{align} A&=A\cap\left((B\cap C)\cup\left(B\cap C\right)^c\right)\\ &=(A\cap B\cap C) \cup \left(A\cap \left(B^c\cup C^c\right)\right)\\ &\subseteq (A\cap B\cap C) \cup \left(B^c\cup C^c\right) \end{align}$$