Thoughts on the empty set and logic

Hello all,

I was just thinking about this today and was wondering, if anyone else had any input on the matter.

As we know the empty set

Now by definition is subset of all sets, because it has no elements or by contradiction because there are not any elements of that are not a member of a set.

so we have

for any A

thus is a tautology.

Yet by using Axiom of extensionality and scheme of comprehension. We can define the empty set = which is a contradiction.

So the definition of an empty set is a contradiction which allows for it to exist, yet its a subset of all sets by tautology.

Is my reasoning here correct, or have i made a logical mistake?