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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 ?

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    Aren't this also implies $A = B$ ?!2010-12-23

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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.

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    @kahen: +1 for NC reference ^_^2011-07-05
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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 $A=B=C$.