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I'm trying to find a counterexample to this statement: "If $A \in B$ and $B \subseteq C$, then $A \in C$."

So I thought I could use $\emptyset$.

Is the empty set a member of a set containing the empty set?

So, if $A = \emptyset $ and $B = \{\emptyset\}$ is $A \in B$?

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    The empty set is a subset of any set. Your statement is actually a true one. – 2017-02-07

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Sure $\emptyset \in \{\emptyset\}$, it's the only member, in fact.

BTW, you wont't find a counterexample to the claim. $B \subseteq C$ is defined as "every member of $B$ is a member of $C$".