This time I've task with minimal sets.

"We say that a structure $\displaystyle A$ is minimal if $\displaystyle A$ is infinite but the only subsets of $\displaystyle \mbox{dom}(A)$ which are first-order definable with parameters are either finite or cofinite (i.e. complements of finite sets). More generally a set $\displaystyle X \subseteq \mbox{dom}(A)$ which is first-order definable with parameters is said to be minimal if $\displaystyle X$ is infinite, and for every set $\displaystyle Z$ which is first-order definable in $\displaystyle A$ with parameters, either $\displaystyle X \cap Z$ or $\displaystyle X\backslash Z$ is finite. "

And now the exercise:

== Let $\displaystyle A$ be the graph whose vertices are all the sets $\displaystyle \{m, n\}$ of exactly two natural numbers, with $\displaystyle a$ joined to $\displaystyle b$ iff $\displaystyle a \cap b \neq \varnothing \wedge a \neq b$. Show that $\displaystyle A$ is not minimal, but it has infinitely many minimal subsets. ==

I'll be grateful if this problem someone can show me step by step. When I understand this exercise I'll do the rest from my list. Because of this I'm asking for details. Thanks.