  We prove a set-theoretic version of the Landsberg--Weyman Conjecture on the
defining equations of the tangential variety of a Segre product of projective
spaces. We introduce and study the concept of exclusive rank. For the proof of
this conjecture we use a connection to the author's previous work
\cite{oeding_pm_paper, oeding_thesis} and re-express the tangential variety as
the variety of principal minors of symmetric matrices that have exclusive rank
no more than one.
