Natural Number Ordering is Transitive
Let $m, n, k \in \N$ where $\N$ is the set of natural numbers.
Let $<$ be the relation defined on $\N$ such that:
- $m < n \iff m \in n$
where $\N$ is defined as the minimal infinite successor set $\omega$.
- $k < m, m < n \implies k < n$
That is: $<$ is a transitive relation.
Let $k < m, m < n$.
By definition it follows that $k \in m, m \in n$.
We have from Element of Finite Ordinal iff Subset that:
- $k \in m \iff k \subseteq m$
- $m \in n \iff m \subseteq n$
It follows from Subset Relation is Transitive that $k \subseteq n$.
Hence the result.