I asked this question [here](https://math.stackexchange.com/q/1332658/75923) 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 :).



Has by well-foundedness every non-empty class an $R$-minimal element? Also if axiom REG is not assumed?
Let me mention another counterexample. In [1, Thm. 11], we construct a model of $\mathrm{ZFC}^-$ with the collection schema which contains a definable class relation $\langle A,<\rangle$ such that


1. $<$ is a dense linear order on $A$ with no least element;
2. every subset of $A$ is well-ordered by $<$.


The second condition says that $<$ is well-founded in the sense used in the question, nevertheless it has no minimal element by 1.


[1] A. S. Daghighi, M. Golshani, J. D. Hamkins, E. Jeřábek, [*The foundation axiom and elementary self-embeddings of the universe*](http://arxiv.org/abs/1311.0814), in: Infinity, Computability, and Metamathematics: Festschrift celebrating the 60th birthdays of Peter Koepke and Philip Welch (S. Geschke, B. Löwe, and P. Schlicht, eds.), College Publications, London, 2014, pp. 89–112.


