Consider the following FOL sentence:

$\phi = \exists x \forall y \exists z ((x=y) \lor (P(x,y,z) \land \lnot P(y,x,z) ) $

It can be proven that for any natural number n > 0 there exits a model of size n for the above sentence. (Please correct me here if I am wrong. This should be provable using induction.).

Now imagine a FOL sentence that **does not use = (and similar) predicate**. And if such a sentence has a model of size n can I claim that the sentence will essentially have a model of size n+1 ?