Equivalency

From chapter 8 of the AI book.

Is the following a  valid (necessary true) sentence?

For all x, either Smart(x) or x = x is true.

Does there exist an x such that x ≠ x (you're welcome for checking on the HTML code for the mathematical "not equals" operator instead of the C-style !=)?  I suppose it depends on our notion of equivalency.  If x is the object that is not equal to anything, not even itself, then this statement is false.  But this self-referencing definition is only constructed for the purpose of creating a paradox, and it is only unique by its own assertion and not by objectivity.  By what seems like a general notion of equivalency, two objects' values can be compared or two objects' identities can be compared.  Certainly the value of x is equal to itself, and so is the identity.  Even if x is indeed the object that is not equal to anything, not even itself, it actually is equal in value and identity to itself no matter what it claims.  So I am going to answer that this is valid.