  We give a new proof of the fact that finite bipartite graphs cannot be
axiomatized by finitely many first-order sentences among FINITE graphs. (This
fact is a consequence of a general theorem proved by L. Ham and M. Jackson, and
the counterpart of this fact for all bipartite graphs in the class of ALL
graphs is a well-known consequence of the compactness theorem.) Also, to
exemplify that our method is applicable in various fields of mathematics, we
prove that neither finite simple groups, nor the ordered sets of
join-irreducible congruences of slim semimodular lattices can be described by
finitely many axioms in the class of FINITE structures. Since a 2007 result of
G. Gr\"atzer and E. Knapp, slim semimodular lattices have constituted the most
intensively studied part of lattice theory and they have already led to results
even in group theory and geometry. In addition to the non-axiomatizability
results mentioned above, we present a new property, called Decomposable Cyclic
Elements Property, of the congruence lattices of slim semimodular lattices.
