I'm stuck on a detail in Luther Claborn's paper Every Abelian Group is a Class Group.
Recall that we say an integral domain $A$ is a Krull domain if
\begin{align*} &\text{(1) }A = \bigcap_{P \in S} A_P, \text{ where $S$ is the set of minimal primes of $A$.} \\ &\text{(2) } A_P \text{ is a DVR for all $P \in S$.} \\ &\text{(3) } \text{For all $a \in A \setminus \{0\}$, $a$ is in only finitely many $P \in S$.} \end{align*} We also say that $\{A_P \,:\, P \in S\}$ is the set of essential valuation rings.
In the proof of Proposition 6, he defines a Krull domain
\begin{align*} B = F[X_1,Y_1,Z_1, \dots, X_i, Y_i, Z_i, \dots ] \end{align*}
($F$ is a field and $i \in J$, where $J$ is an arbitrary index set), and
\begin{align*} R_i = F[\dots,X_i,Y_i,W_i,\dots] \end{align*}
where $W_i = X_iZ_i$. Form the ideals $(X_i,Y_i)$ in $R_i$, from which we get some valuations $v_i(r) = t$ if $r \in Q_i^t$ and $r \notin Q_i^{t+1}$. These can be extended to the quotient field of $R_i$ which is the same as the quotient field for $B$, and we denote $V_i$ to be the corresponding valuation rings. Now, $B$ is a UFD, so $B = \bigcap_P B_P$, where $P$ ranges over all minimal primes of $B$. The goal is to show that $A = \left(\bigcap_i V_i \right)\cap B$ is a Krull domain with essential valuation rings $\Sigma = \{V_i\} \cup \{B_P\}$.
I can see that $A = \bigcap_i V_i \cap \bigcap_P B_P$ and that the elements of $A$ can only appear in finitely many of the minimal primes corresponding to the rings in $\Sigma$. I don't see why $\Sigma$ is the set of essential valuation rings. In the paper, Claborn claims:
"To see this, we need only produce an element of the quotient field which is not in a particular ring of $\Sigma$ but is in all other rings of $\Sigma$."
In my mind, we need to check that if $P_0$ is a minimal prime such that $A_{P_0} \notin \Sigma$, then somehow the claim above should contradict the minimality of $P_0$. Is this right? Also, is showing the claim for a given set of valuation rings enough to show that set is essential in any Krull domain, or is it just true in this specific example?
Thanks in advance for any help!