Elementary Set Theory – How to Prove the Empty Set is Well Ordered

Question

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?

Answer 1

Apply the definition :

$\emptyset$ is well-ordered if it has a total order and every non-empty subset of $\emptyset$ has a least element in this ordering.

It means :

$\forall S (S \subseteq \emptyset \land S \ne \emptyset \to \ldots)$.

But the only subset of $\emptyset$ is $\emptyset$ itself.

Thus, $S \subseteq \emptyset \land S \ne \emptyset$ is False, and $(S \subseteq \emptyset \land S \ne \emptyset \to \ldots)$ is True, for every $S$.