Let $\mathscr{L}$ be a first-order language with two unary predicates $A^1_1$ and $A^1_2$. To have an interpretation $M$ of $\mathscr{L}$ that makes the well-formed formula (wf)
$$(\exists x₁)A^1_1(x₁) ∧ (\exists x₁)\neg A^1_1(x₁) $$
true, it seems the only condition would be that the domain of $M$ has at least two elements. Now I am trying to find a wf that can only be true if the domain of the interpretation has at least four elements. Would the following do the trick?
$$(\exists x₁)A^1_1(x₁) ∧ (\exists x₁)\neg A^1_1(x₁) ∧ (\exists x_2)A^1_2(x_2) ∧ (\exists x₂)\neg A^1_2(x_2) $$
No that wouldn't do the trick because it could be the same thing that e.g. both satisfies $A_1$ and $A_2$. So that wff is still satisfiable in any domain with at least two elements.
If you want to continue avoiding using the identity predicate, how about trying a conjunction with various clauses of the form $\exists x(?A_1x\ \land\ ?A_2x)$ where the ? is either empty or the negation sign ...