I tried to prove this statement:
$$[(A\cup B)\cap C = A\cup(B\cap C)]\iff A\subset C$$
I did it in the following way, can anyone tell me if it's correct what I've done?
$\leftarrow$ Assume $A\subset C$, then $\forall x \in A$, $x\in C$
Then, $\forall x \in (A\cup B)\cap C$, $x\in C$ and $\in B$
Similarly, for $\forall x \in A\cup(B\cap C)$, $x\in B$ and $x\in C$
So $(A\cup B)\cap C = A\cup(B\cap C)$
$\rightarrow$ I didn't know how to do the counterpart.
Could someone please help me out?