TITLE: $A \in B$ vs. $A \subset B$ for proofs
QUESTION [12 upvotes]: 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 $A \in C$.

REPLY [2 votes]: You can't prove that if $A \in B$ and $B \in C$ then $A \in C$ because it is not always true.
Let $B = \{A, 1 \}$. Then $A \in B$
Let $C = \{ B,2\} = \{ \{A, 1 \}, 2 \}$
The members of $C$ are $\{A, 1 \}$ and $2$. So $A \notin C$.
Yes $A$, is in $\{A, 1 \}$, but it is not in $C$.