Every nonempty finite set $x$ has an $\in$-maximal element $y\in x$, since one can simply climb up via $\in$ inside $x$ until one reaches a maximal element.
But the statement is not true of all infinite sets, since the usual ordinal $\omega$, for example, is closed under the singleton operation and therefore has no $\in$-maximal elements.
Therefore, the property $\varphi(x)$ asserting: $$x\neq\emptyset\to \exists y\in x\ \forall z\in x\ (y\notin z)$$ is true of every finite set, but it does not transfer to the infinite realm.
Since this assertion is very short, it would seem to be a violation of your transfer principles. But I'm not actually sure, since I haven't followed your axioms and notation in detail.