elementary set theory - $(A\cap B)\cup C = A \cap (B\cup C)$ if

I have a set identity: $(A \cap B) \cup C = A \cap (B \cup C)$ if and only if $C \subset A$. I started with Venn diagrams and here is the result: It is evident that set identity is correct. So I

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The ( left( A cap B ^ { prime } right) ^ { prime } cup ( B cap C

The set $\left( {A \cup B \cup C} \right) \cap \left( {A

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