Minimal Element/Examples/Finite Subsets of Natural Numbers

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

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

There is one minimal element of $\struct {\FF, \subseteq}$, and that is the empty set $\O$.

Proof
We have that $\O$ is a finite set.

By Empty Set is Subset of All Sets it follows that $\O$ is a subset of $\N$.

Hence $\O \in \FF$ by definition of $\FF$.

Let $A \in \FF$ be some finite subset of $\N$ such that $A \subseteq \O$.

Then by Subset of Empty Set iff Empty:
 * $A = \O$

Hence $\O$ is a minimal element of $\struct {\FF, \subseteq}$ by definition.