Following question: Let's assume that W is a wellfounded set, i.e. it has a partial order and every nonempty subset of W has minimal elements with respect to the order.

Now we can easily define a binary relation 'preceeds' with the definition

a.preceeds(b) = b.is_minimal({x: a < x})

I am not able to prove that the fact that an element b has no predecessor (with respect to the preceeds relation) implies that b is minimal in W.

Is it possible in a wellfounded set that an element b has no predecessor but there are elements a below it (i.e. a < b)? If this is possible are there examples?

Thanks for any help.