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Suppose $A \subseteq C$ and $B$ and $C$ are disjoint. Prove that $x \in A \rightarrow x \notin B$.

Basically I need to prove this.

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Not much to it: Since $A\subset C$, $a\in A\Rightarrow a\in C$. Since $C\cap B=\emptyset$, $a\in C\Rightarrow a\notin B$. – David Mitra Jan 30 '12 at 21:36
Since $x\in A$ and $A\subseteq C$ then ...; but $B$ and $C$ are disjoint, so ... – Andrés E. Caicedo Jan 30 '12 at 21:36

In proving these things you need to show that no matter how $A,B,C$ look like, if the condition that $A\subseteq C$ and $B\cap C=\varnothing$ then $x\in A\rightarrow x\notin B$ is true.

For this we want to take an arbitrary element of $A$, use the definition that $A\subseteq C$ to deduce more about $x$; and use the definition of $B\cap C$ to deduce that if $x$ was in $B$ then the intersection would not be empty - therefore $x\notin B$.

I leave the formal and technical details to you, since this is your homework assignment after all.

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