Equivalence of Definitions of Artinian Module

Definition 1 iff Definition 2
Let $D$ be the set of all submodules of $M$.

We shall show that:
 * descending chain condition
 * minimal condition

with respect to $\struct {D, \supseteq}$ are equivalent.

This is nothing but:
 * ascending chain condition
 * maximal condition

with respect to $\struct {D, \subseteq}$ are equivalent.

The latter follows from Increasing Sequence in Ordered Set Terminates iff Maximal Element.