I'm trying to understand a proof on "Sheaves of Continuous Definable Functions" (Pillay, Anand. "Sheaves of continuous definable functions." The Journal of symbolic logic 53.04 (1988): 1165-1169.)
Let $\mathcal{N} = (N,<,\ldots)$ be an o-minimal structure and let $X \subset N^m$ be a definable set, the o-minimal spectrum $X^\sim$ of $X$ is the set of complete m-types $S_m(N)$ of the first-order theory $Th_N (\mathcal{N})$ which imply a formula defining X equipped with the topology generated by the basic open sets of the form $$U^\sim = \{p \in X^\sim : U ∈ p\}$$ where U is a definable, relatively open subset of X. We call this topology on $X^\sim$ the spectral topology.
My goal is to understand why is $X^\sim$ a spectral space. Let $F \subset X^\sim$ be a closed, irreducible space. The proof goes along the lines: Let $\Phi = \{B \subset X : B $ is closed, definable and $B \in p$ for every $p \in F\}$ and then $\Phi'= \{ X \setminus C: C$ closed, definable and $C \notin \Phi\}$, and $\Phi_1 = \Phi \cup \Phi'$ and argues that, since $F$ is irreducible, $\Phi_1$ is consistent.
I already know that $\Phi$ is consistent (since $\Phi \subset p$ for every $p \in F$) and that for every $\varphi \in \Phi'$ either $\Phi \cup \varphi$ or $\Phi \cup \lnot \varphi$ is consistent. But I can't relate that to $F$ irreducibleness.