The converse of the previous proposition is not true in general, for instance, the ordered set of integers $(\mathbb Z,\ge)$ is a chain, but it is not well-ordered, since not every of it subsets contains a minimum, e.g. the infinite subset $\{-1,-2,-3,\ldots \}.$ It is only true, if $V$ is finite:
Every finite chain $V$ is well-ordered.
Proofs: 1
