With our normal axioms of set theory is it proper to say that an undefined value is not an element of a set containing all defined values?
As an example we could be asking "Is the final digit in the decimal expansion $0.101010...$ a member of the set $\{0,1\}$?" Here the final digit is not defined but saying "the final digit is neither $0$ nor $1$" is in one way having some defined final digit to compare with $0$ and $1$.
What this boils down to is does it make sense to compare an undefined value to many defined values?
There's no such thing as an 'undefined value', because if a value is undefined, then it isn't a value. When we say something is undefined, what we really mean is either that a written expression has no meaning, or that there are no objects with a certain property.
Here, 'the last digit of $0.101010\dots{}$' is undefined, in the sense that $0.101010\dots$ has no last digit. So when you ask if something is true of the last digit of $0.101010\dots{}$, you're asking if a property holds of an element of the empty set... which is not a very fruitful business.