$$A \, \text{never equal to} \, B \implies \, \forall\, \text{valid values of } (A, B)\, , A \text{ is different in value from }B$$
As opposed to $A \neq B$.
$$A \neq B \implies \exists\, \text{ a valid } (A, B)\text{: } A \, \text{is different in value from }B$$
Is there a symbol to conveniently represent the former.
$0$ is both not equal to $1$ and never equal to one. I may be given two functions $f(x)$ and $g(x)$. $f$ and $g$ may produce different results for some $x$ (not equal), always produce the same result (equal) or never produce the same result (never equal).
$A=f$, and $B = g$.
The set of "not equal to" is a proper superset of the set of "never equal to", and both are disjoint with the set of "equal to".
Best Answer
If we follow your definition(s), yes there is a symbol for that: $F$.
Indeed,
$$\forall A, B\in D:A\ne B$$ is false (take $A=B=C$ where $C\in D$ as a counterexample).
Assuming anyway that $D\ne\emptyset$.
The predicate
$$\exists A, B\in D:A\ne B$$ is false for empty sets and singletons, and true otherwise. Hence it can be written $\#D>1$.
Update:
Speaking of the valid values of the pairs $(A,B)$ implies that you are talking about a binary relation, i.e. a set of pairs. Then you are looking for a way to express that the relation is irreflexive. Such a predicate will involve the relation, not a particular pair $(A,B)$.
I have never seen a specific shorthand, you might write $\text{Irreflexive(R)}$ where $R$ denotes the relation.