I asked this question here on Math.SE but uptil now it was not answered. So I decided to give it a try. Thank you in advance.
Working in $\mathbf{ZF}$ let $R$ be a proper class of ordered pairs that is well-founded. This means that for every non-empty set $a$ there is a set $b\in a$ such that $cRb\implies c\notin a$. Here $cRb$ is a notation for $\langle c,b\rangle\in R$ and $b$ is a so-called $R$-minimal element of $a$. If $R$ is local (i.e. collections $\{x\mid xRb\}$ are all sets) then it can be shown that also non-empty proper classes have $R$-minimal elements.
I encountered a proof that made the condition of being local redundant. It made use of an operation on classes that adds to each class a set that is contained in it (Bottom-operation) but the definition of this operation relied on the regularity axiom.
My question:
Is there a proof that every non-empty class has an $R$-minimal element that does not make use of the regularity axiom?
Another formulation:
If $R$ is a class of ordered pairs that is well-founded, then is it legal to apply $R$-induction if all axioms of $\mathbf{ZF}$ are accepted with exception of the axiom of regularity?
Edit:
Maybe a bounty will help. If the question will not be answered then I will start cherishing the fact that I asked a good question ($6$ upvotes) that 'nobody' could answer :).
