When was Regularity/Foundation universally adopted?
Every modern presentation of ZF includes the axiom of regularity (or foundation): every nonempty set $x$ has an element $y$ with $x\cap y=\emptyset$. Of course, ZF is a system which developed over time, and going back far enough we can find various different versions and predecessors of the modern system.
That said, I was quite surprised to find out recently that in Mendelson's text on logic states that the axiom of regularity is not necessarily a standard one:
In recent years, ZF is often assumed to contain [the axiom of regularity]. The reader should always check whether [regularity] is included in ZF. $\quad$(pg. 288)
Moreover, a review by Van Dalen doesn't mention this, suggesting that Mendelson's commentary on the axiom isn't out of place. This is all especially surprising to me since I was under the impression that it was adopted in the 30s.
My question is:
When did regularity become universally accepted as an axiom of ZF?
Of course I'm making an assumption here - that regularity is universally accepted as an axiom of ZF in the modern era. While it's impossible to prove that there isn't some recently written text somewhere which doesn't include it, in lieu of an example of such a text (say, from the last 40 years) I think I'm more than justified in making this claim; e.g. every major textbook from the last 40 years of which I'm aware (Bar Hillel/Fraenkel/Levy/Van Dalen, Kunen, Jech, Enderton, Ciesielski,...).
Solution 1:
Jean Louis Krivine's book Introduction to Axiomatic Set Theory (1971) defines the theory ZF (chapter 1) as not containing the Axiom of Foundation (introduced in chapter 3). This edition is the English translation of the French version of this book, published in 1969.
The same author published a new book Theorie des ensembles (2007, only in French) keeping such a definition of ZF.
In my opinion, the fact of non including the Axiom of Foundation in the definition of ZF is quite popular at least in France. I don't know if this is due to Krivine's first book, or if Krivine was following an already consolidated French tradition. Bourbaki's Théorie des ensembles (1970, only in French) does not mention the Axiom of Foundation. Anyway, this definition of ZF is still valid today in France. For instance, Cori's and Lascar's undergraduate textbook Mathematical Logic: A Course With Exercises, Part II (2001, largely used in courses of mathematical logic in French universities) follows this definition in chapter 7.