What is the Identity for Unions and Intersections of Sets?

[Dustinsfl]

$(A\cup B)\cap (B\cup C)\cap (C\cup A) = (A\cap B)\cup (A\cap C)\cup (B\cap C)$

For the identity, we will show $(A\cup B)\cap (B\cup C)\cap (C\cup A) \subseteq (A\cap B)\cup (A\cap C)\cup (B\cap C)$ and $(A\cup B)\cap (B\cup C)\cap (C\cup A) \supseteq (A\cap B)\cup (A\cap C)\cup (B\cap C)$.
Let $x\in (A\cup B)\cap (B\cup C)\cap (C\cup A)$.
Then $x\in A\cup B$ and $x\in B\cup C$ and $x\in C\cup A$.
So $x\in A$ or $x\in B$ and $x\in B$ or $x\in C$ and $x\in C$ or $x\in A$.

So I am stuck at this point.

 

[Fantini]

If ($x \in A$ or $x \in B$) and ($x \in B$ or $x \in C$) and ($x \in C$ or $x \in A$), then ($x \in A$ and $x \in B$) or ($x \in A$ and $x \in C$) or ($x \in B$ and $x \in C$). From there, $x \in (A \cap B) \cup (A \cap C) \cup (B \cap C)$.