Corollary

Every intersection of a non-empty family of integrally closed subdomains of an integral domain $A$ is an integrally closed domain.

It is sufficient to apply Proposition 8 taking $R$ and the $B_\lambda$ equal to the field of fractions $K$ of $A$ and noting that a subring of $K$ integrally closed in $K$ is a fortiori an integrally closed domain since its field of fractions is contained in $K$.