where $2^N$ is the power set with $n$ elements (subsets).
Does it hold true to any set or just the power set $2^N$?
By definition, two sets $A$ and $B$ are equal if and only if $A\subseteq B$ and $B\subseteq A$. Now a relation $R$ on a set $X$ is antisymmetric if $aRb$ and $bRa$ implies $a=b$ for all $a,b\in X$. Does this help you see why $\subseteq$ is antisymmetric?