$A \in B$ vs. $A \subset B$ for proofs I have to prove a few different statements. 
The first is if $A \subset B$ and $B \subset C$ then prove $A \subset C$. This one is fairly straight forward, but I'm stuck on how the next one differs. 
Prove that if $A \in B$ and $B \in C$ then $A \in C$. 
I don't really understand how to put this in logical symbols. I've only seen $a \in A$ written out but never "a set $A$ is an element of a set $B$".
Here's what I have for a proof at this point, assuming I understand what "a set $A$ is an element of a set $B$" means: suppose $A \in B$ and $B \in C$. 
Then $A \in C$.

Take $A=\varnothing$, $B=\{\varnothing\}$ and $C=\{\{\varnothing\}\}$.
Then clearly $A\in B\wedge B\in C$, but $A\in C$ implies $\varnothing=\{\varnothing\}$ wich cannot be true. 
This because $\{\varnothing\}$ has elements and $\varnothing$ has not.
We conclude that the implication is false.
