Well-founded and setlike relations

Let $X$ be a definable class and $R$ a definable binary relation on $X$, with fixed set parameters. Write ${\mathrm{pred}}_R(x)=\{y\in X:yRx\}$. The relation is setlike if this predecessor collection is a set for every $x\in X$. It is well-founded if every nonempty set $A\subseteq X$ has an $R$-minimal member $x$, meaning ${\mathrm{pred}}_R(x)\cap A=\mathrm{\varnothing }$.

These are schemes in first-order set theory: a class is notation for a defining formula. No transitivity or totality of $R$ is required. Every relation on a set is setlike. An ordinal carries a well-founded membership relation by its definition; ambient Foundation does not make every arbitrary relation well-founded. All results concerning a supplied well-founded setlike relation are valid in ZF without Foundation unless stated otherwise.

Conventions and prerequisites: Ordinal (von Neumann).