Proper notation for distinct sets

No, there is no special symbol to denote the situation in which neither set is contained in the other. There is a special word, though: we say the two sets are incomparable.

There is also no special symbol to denote the fact that two sets are disjoint; we simply write $A\cap B=\emptyset$.

Let $\rm\:P\: $ be a poset = partially-ordered set, i.e. a set equipped wth a relation $\rm\le$ that is reflexive, antisymmetric and transitive. Then one says that $\rm\: x\: $ and $\rm\: y\: $ are comparable$\ $ if $\rm\ x \le y\ $ or $\rm\ y \le x\:;\: $ otherwise they are incomparable, sometimes written $\rm\ x||y\:.\:$ A chain$\ $ of $\rm\: P\:$ is a subset in which every two elements are comparable, and dually, an antichain$\ $ is a subset in which every two elements are incomparable.

Not an answer, perhaps an interesting historical curiousity. George Mackey (my professor years ago) hoped to popularize this symbol as a binary relation for disjoint sets:

I think he meant it as a lower case "d" drawn symmetrically.

The screen shot suggests that $\TeX$ doesn't know it.

Edit: This question on $\TeX$ stackexchange provides an update: https://tex.stackexchange.com/questions/554944/looking-for-a-disjoint-symbol