# Compliments Proof 3

Let $X \subset Y \subset Z$. Prove that $C_{Y}(X) \subset C_{Z}(X)$.

In order for $C_{Y}(X) \subset C_{Z}(X), \forall_{x \in C_{Y}(X)}(x \in C_{Z}(X)) \Rightarrow \exists_{x \notin X} : x \in Y \cup Z$ because $X \subset Y \subset Z$. If we assume that $\forall_{x \in X}(x \in Y \cup Z)$ then $C_{Y}(X)$ and $C_{Z}(X)$ would both be $\emptyset$ and $\emptyset \not\subset \emptyset$. $\therefore \exists_{x \notin X} : x \in Y \cup Z$. Also $\not\exists_{x \in X}: x \notin Y \cup Z$ because then $X \not\subset Y$ which creates a contradiction with the axiom above. Since $\exists_{x \in Y \cup Z}: x \notin X$, $\forall_{x \in C_{Y}(X)}(x \in C_{Z}(X)) \Rightarrow C_{Y}(X) \subset C_{Z}(X)$.