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 $\displaystyle \{\} = \theta$

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

so we have

$\displaystyle \theta \sqsubseteq A $ for any A

thus $\displaystyle \theta \sqsubseteq A $ is a tautology.

Yet by using Axiom of extensionality and scheme of comprehension. We can define the empty set $\displaystyle \theta$ = $\displaystyle \{x \in A | x \not= x \} $ 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?