Smallest 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 $\struct {\FF, \subseteq}$.

Then $\struct {\FF, \subseteq}$ has a smallest element, 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 $\N$.

Then by Empty Set is Subset of All Sets:
 * $\O \subseteq A$

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