I found a post concerning this question, but I cannot understand the proof given there of the fact I'm talking about.
The link is : Verification of proof that the empty set is well ordered
I think I can prove indirectly that the empty set is well ordered in the following way :
(1) suppose the empty set is not well ordered
(2) that is, suppose it is false that every non empty subset of the empty set has a first element
(3) it means there exists at least some set S such that
S is a non empty subset of the empty set
and S has no first element
(4) which requires the first conjunct " S is a non empty subset of the empty set" to be true
(5) but this is impossible, for the empty set has only one subset, which is empty.
However , I cannot manage to give a direct proof of the same fact.
I cannot go further than this : (1) Let S be an arbitrary set (2) Assume it is true that : S is a non empty subset of the empty set (3) Derive from this that : S has a first element. ... but how?