this is an exercise from How to solve it by Daniel J. Velleman. My proof differs from the one given in the book and I was wondering if it is still valid.
Suppose $A \cap C \subseteq B \cap C$ and $A \cup C \subseteq B \cup C$. Prove that $A \subseteq B$.
My proof: Let $x$ be arbitrary. Suppose $x \in A$. Since $A \cup C \subseteq B \cup C$, then $x \in B \cup C$. Then we have the following two cases.
Case 1: Suppose $x \in B$. This is our goal.
Case 2: Suppose $x \in C$. Since $A \cap C \subseteq B \cap C$, then $x \in B \cap C$, which means $x \in B \land x \in C $, which means $x \in B$.
Since x was arbitrary then $\forall x: x \in A \rightarrow x \in B$, which means $A \subseteq B$.
I don't see any logical errors in my proof, but I'd be really grateful if somebody could check it.