Basic Facts about Intervals of the Real Line

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let intervals, the bounded intervals $[a,b]$, $(a,b)$, $[a,b)$, $(a,b]$ and the rays $(a,\infty)$, $[a,\infty)$, $(-\infty,b)$, $(-\infty,b]$ be as in Intervals of the Real Line. Let $I\subseteq\mathbb{R}$ be an interval and let $a,b,p,q,u,v,x\in\mathbb{R}$.

(i) $\mathbb{R}$ is an interval containing two distinct points, and there are $s,t\in\mathbb{R}$ with $s<x<t$.

(ii) If $u,v\in I$ and $u\le v$, then $[u,v]\subseteq I$.

(iii) If $p,q\in I$ and $p<x<q$, then $x\in I$.

(iv) $(p,q)$ is an interval, and if $x\in(p,q)$, then there are $s,t\in(p,q)$ with $s<x<t$.

(v) If $a\le b$, then $[a,b]$ is an interval, $a,b\in[a,b]$ and $(a,b)\subseteq[a,b]$; if moreover $a<b$, then $[a,b]$ contains two distinct points.

(vi) $[a,b]$, $(a,b)$, $[a,b)$, $(a,b]$, $(a,\infty)$, $[a,\infty)$, $(-\infty,b)$ and $(-\infty,b]$ are intervals.

(vii) If $I$ contains two distinct points, $x\in I$, $\delta\in\mathbb{R}$ and $\delta>0$, then there is $t\in I$ with $0<|t-x|<\delta$.