In the chapter on localization in Neukirch, Algebraic Number Theory, the following is mentioned:
(A is an integral domain) Usually $S$ will be the complement of a union $\bigcup_{\mathfrak p \in X}\mathfrak p$ over a set $X$ of prime ideals of $A$. In this case one writes $$A(X) = \{\frac fg \mid f,g \in A, g \not \equiv 0 \bmod{\mathfrak p}\text{ for }\mathfrak p\in X\}$$ The prime ideals of $A(X)$ correspond $1-1$ to the prime ideals of $A$ which are contained in $\bigcup_{\mathfrak p \in X}\mathfrak p$. For instance if $X$ is finite or omits only finitely many prime ideals of $A$, then only the prime ideals from $X$ survive in $A(X)$.
The passage seems to say that the only prime ideals contained in $\bigcup_{\mathfrak p \in X}\mathfrak p$ are those in $X$ but is this true in general? What if there is some prime ideal contained in one of the ideals of $X$?