I have an irreflexive and transitive relation $R$. Then I want to prove that $\forall x \exists y (xRy)$ has only infinite models. I have an intuitive idea for which the relation $R$ cannot be reflexive (cause transitivity + symmetry implies reflexivity), then we know that everything is not related to everything. In this sense we cannot have a closed circle of relation and clearly there it must be an element of the domain such that there is nothing else that is related to it. We could then add another element but the same reasoning still apply.
How can I prove this formally?