I am trying to prove that the DLO $(\mathbb{Q}, <)$ is an $O$-minimal structure, but I am trying to prove this without using the fact that the complete theory of dense linear orderings without endpoints admits quantifier elimination.
Here is my proof thus far: Let $N = \varphi(\mathbb{Q}, \bar{a})$ where $\varphi$ is a first-order formula from the language of partial orders. We may assume that $\bar{a} = (a_0, \ldots, a_{n-1})$ and that $a_0<\cdots
Assumption: for any $-1\leq j\leq n-1$ and $x,y \in \mathbb{Q}$ such that $a_j
If this is true, then if $N$ contains an element of the interval $(a_j, a_{j+1})$ it must then contain the whole open interval, and from this we can infer that the structure is O-minimal.
I am having difficulty, however, proving the assumption. Any help would be greatly appreciated.