Minimal Element/Examples/Finite Subsets of Natural Numbers less Empty Set

Examples of Minimal Elements
Let $\FF$ denote the set of finite subsets of the natural numbers $\N$.

Let $\GG$ denote the set $\FF \setminus \set \O$, that is, $\FF$ with the empty set excluded.

Consider the ordered set $\struct {\GG, \subseteq}$.

The minimal elements of $\struct {\GG, \subseteq}$ are the sets of the form $\set n$, for $n \in \N$.

Proof
Let $n \in \N$.

We have that $\set n$ is a finite subset of $\N$.

Hence $\set n \in \GG$ by definition of $\GG$.

Let $A \in \GG$ be some finite subset of $\N$ such that $A \subseteq \set n$.

Then either $A = \O$ or $A = \set n$.

Because $\O \notin \GG$ it follows that $A = \set n$

Hence $\set n$ is a minimal element of $\struct {\GG, \subseteq}$ by definition.

Now suppose $B \subseteq \N$ such that:
 * $\exists m, n \in B: m \ne n$

Then $B$ is a finite subset of $\N$ such that $B \ne \O$.

Hence $B \in \GG$ by definition of $\GG$.

Note that we have:
 * $\set n \subseteq B$

and also:
 * $\set m \subseteq B$

but $B \ne \set n$ and $B \ne \set m$.

Hence $B$ is not a minimal element of $\struct {\GG, \subseteq}$.