  We study properties of metric segments in the class of all metric spaces
considered up to an isometry, endowed with Gromov--Hausdorff distance. On the
isometry classes of all compact metric spaces, the Gromov-Hausdorff distance is
a metric. A metric segment is a class that consists of points lying between two
given ones. By von Neumann--Bernays--G\"odel (NBG) axiomatic set theory, a
proper class is a "monster collection", e.g., the collection of all cardinal
sets. We prove that any metric segment in the proper class of isometry classes
of all metric spaces with the Gromov-Hausdorff distance is a proper class if
the segment contains at least one metric space at positive distances from the
segment endpoints. In addition, we show that the restriction of a
non-degenerated metric segment to compact metric spaces is a non-compact set.
