how does $(A \cap B) \cap (B \cap C)$ lead to $A \cap B \cap C$?

Question

I was reading the proof of inclusion-exclusion principle for three events. I am wondering how does $$(A\cap B)\cap(B\cap C)$$ lead to $A\cap B \cap C$? Is it by associativity so that we can rewrite as $A \cap (B \cap B) \cap C$ so then becomes $A \cap B \cap C$?

Answer 1

Made it just quick and easy. You see what the green and pink subset have in common.

Answer 2

Yes, that is one way to show the relation. Another way is to check that $x\not\in S$ for each $S\in\{A,B,C\}$ implies $x\not\in A\cap B\cap C$.