Suslin lines in order language

A Suslin line here is a linearly ordered set $(L,<)$ with the following properties. Linear means that the associated reflexive order is a partial order as in Partial order and partially ordered set and every two elements are comparable.

It is nonempty, dense ($x<y$ implies some $z$ satisfies $x<z<y$), and has no endpoints (for every $x$ there are $u<x<v$). It is Dedekind complete: every nonempty subset $B\subseteq L$ that is bounded above has a least upper bound in $L$. A subset $D\subseteq L$ is order-dense if it meets every open interval $(x,y)=\{z:x<z<y\}$ for $x<y$; $L$ has no countable order-dense subset, where countable includes finite as in Finite, countably infinite, countable, uncountable. Finally, every family of pairwise disjoint nonempty open intervals is countable.

These conditions describe nonseparability of the whole line; they do not require every interval to be nonseparable. A reduction to a nowhere-separable line is a separate result. Completeness requires upper bounds only for nonempty bounded subsets, so it neither asks for a supremum of the empty set nor supplies endpoints. Empty and singleton orders are excluded by nonemptiness and the no-endpoint condition respectively. Every interval with distinct ordered endpoints is nonempty by density.