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

Complement (set theory) - Wikiwand

1.6: Set Operations with Three Sets - Mathematics LibreTexts

DM4CS Methods of Proof for Sets

Sets and Important Notations

SOLVED: For all sets A and B, prove via an element argument that

theorems of probability, properties of probability