Elements of a Segment Are Natural Numbers, and Mixed Transitivity of the Order on the Natural Numbers

In the setting of The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion, let $\le$ and $<$ be the order and the strict order of $\mathbb{N}_{0}$, as in The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion (v), and let $[n]$ be as in Intervals of Natural Numbers (ii).

(i) Let $n$ be a set, not necessarily an element of $\mathbb{N}_{0}$. Every $k\in\mathbb{N}_{0}$ with $1\le k\le n$ is a natural number, $k\in\mathbb{N}$; only the lower bound $1\le k$ is needed for this, the upper bound $k\le n$ plays no role. In particular, for $n\in\mathbb{N}_{0}$, every element of $[n]$ is a natural number.

(ii) Let $k,l,m\in\mathbb{N}$. If $k<l$ and $l\le m$, then $k<m$; and if $k\le l$ and $l<m$, then $k<m$.