  Jeroslow and Lowe gave an exact geometric characterization of subsets of
$\mathbb{R}^n$ that are projections of mixed-integer linear sets, a.k.a
MILP-representable sets. We give an alternate algebraic characterization by
showing that a set is MILP-representable {\em if and only if} the set can be
described as the intersection of finitely many {\em affine Chv\'atal
inequalities}. These inequalities are a modification of a concept introduced by
Blair and Jeroslow. This gives a sequential variable elimination scheme that,
when applied to the MILP representation of a set, explicitly gives the affine
Chv\'atal inequalities characterizing the set. This is related to the
elimination scheme of Wiliams and Williams-Hooker, who describe projections of
integer sets using \emph{disjunctions} of Chv\'atal systems. Our scheme extends
their work in two ways. First, we show that disjunctions are unnecessary, by
showing how to find the affine Chv\'atal inequalities that cannot be discovered
by the Williams-Hooker scheme. Second, disjunctions of Chv\'atal systems can
give sets that are \emph{not} projections of mixed-integer linear sets; so the
Williams-Hooker approach does not give an exact characterization of MILP
representability.
