How is the relation $R$ on the power set of $A$ such that $(x,y) \in R$ if and only if $x \subset y$ anti-symmetric?
I am having trouble understanding how this relation is anti-symmetric. I understand that it is not symmetric as $x$ can be a proper subset of $y$ but that same set $y$ cannot be a proper subset of $x$. How is it that this relation is anti-symmetric. I am having trouble finding an example that would prove this and that ultimately if $(x,y)$ is in the relation and $(y,x)$ is in the relation, then $x = y$.
If you have $(x,y) \in R$ then $x \subseteq y$. If you have $(y,x) \in R$, then $y \subseteq x$. Finally, if you have both, you must have $x \subseteq y \subseteq x$, which implies $x = y$.