How to prove :: $ A \subset B, B \subset C, C \subset A \Rightarrow B = C $ Let $A,B \text{ and } C$ are three sets then if $ A \subset B, B \subset C, C \subset A \Rightarrow B = C $
How could we prove this ?
 A: This is effectively asking to prove $B \subseteq C \land C \subseteq B \implies B = C$. The usual way to prove this is to use the Axiom of Extensionality - i.e. take an element $b \in B$ and show that it is in $C$. Then show that $c \in C \implies c \in B$. Extensionality now tells you that the two sets are identical. 
A: Verbosely:
Say that $A \subseteq B \subseteq C \subseteq A$. Then in particular,
$x \in A \Rightarrow x \in B$ by the first inclusion but then by the second we have $x \in B \Rightarrow x \in C$.
Contracting, $x \in A \Rightarrow x \in C$, but the rightmost inclusion tells us that $x \in C \Rightarrow x \in A$ so that $x \in A \Leftrightarrow x \in C$. By the axiom of extensionality we obtain that $A = C$.
Now by the second inclusion, $x \in B \Rightarrow x \in C$, but since $C=A$, we must have $x \in B \Rightarrow x \in A$, so that with the first inclusion $x \in A \Leftrightarrow x \in B$ and again by the axiom of extensionality we have that $A=B$. Now $A=C$, and so $A=B=C$.
