Maximal 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 $\FF \setminus \O$, that is, $\FF$ with the empty set excluded.

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

There are no maximal elements of $\struct {\GG, \subseteq}$.

Proof
$A \in \GG$ is a maximal element of $\struct {\GG, \subseteq}$.

Then $A$ consists of a finite number of natural numbers.

Let $M = \map \max A$ denote the maximum of the elements of $A$.

Now consider the set:
 * $A' = A \cup \set {M + 1}$

We have that $A'$ is a finite subset of $\N$.

Hence $A' \in \GG$.

But also from Set is Subset of Union:
 * $A \subseteq A'$

while because $M + 1 \notin A$:
 * $A' \ne A$

Hence $A$ cannot be a maximal element of $\struct {\GG, \subseteq}$.

The result follows from Proof by Contradiction.