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日本語の情報が少ないが
重要ポイントで、ur-element ＝ 原始要素 を許容するか否かが論点としてある
純ZFCは、ur-element を許容しない（下記）
しかし、日本の学部で１年次に教える集合論は、だいたいが ur-element を許容しているはず
これで、ZFCの位置づけが 理解できる
即ち、ur-element （＝ 原始要素）を許容しないと メンドクサイし
ur-element を許容しても、同じ議論が ZFCの議論に翻訳できるってことです
下記 ”Urelements in set theory
The Zermelo set theory of 1908 included urelements, and hence is a version now called ZFA or ZFCA (i.e. ZFA with axiom of choice).[1] It was soon realized that in the context of this and closely related axiomatic set theories, the urelements were not needed because they can easily be modeled in a set theory without urelements.[2]”
ってこと
つまり、基礎論としては ur-elementなしがすっきり
しかし、基礎論の外では ur-elementありがすっきりｗ

(参考)
https://en.wikipedia.org/wiki/Urelement
In set theory, a branch of mathematics, an urelement or ur-element (from the German prefix ur-, 'primordial') is an object that is not a set (has no elements), but that may be an element of a set.
Urelements in set theory
The Zermelo set theory of 1908 included urelements, and hence is a version now called ZFA or ZFCA (i.e. ZFA with axiom of choice).[1] It was soon realized that in the context of this and closely related axiomatic set theories, the urelements were not needed because they can easily be modeled in a set theory without urelements.[2] Thus, standard expositions of the canonical axiomatic set theories ZF and ZFC do not mention urelements (for an exception, see Suppes[3]). Axiomatizations of set theory that do invoke urelements include Kripke–Platek set theory with urelements and the variant of Von Neumann–Bernays–Gödel set theory described by Mendelson.[4] In type theory, an object of type 0 can be called an urelement; hence the name "atom".
Adding urelements to the system New Foundations (NF) to produce NFU has surprising consequences. In particular, Jensen proved[5] the consistency of NFU relative to Peano arithmetic; meanwhile, the consistency of NF relative to anything remains an open problem, pending verification of Holmes's proof of its consistency relative to ZF. Moreover, NFU remains relatively consistent when augmented with an axiom of infinity and the axiom of choice. Meanwhile, the negation of the axiom of choice is, curiously, an NF theorem. Holmes (1998) takes these facts as evidence that NFU is a more successful foundation for mathematics than NF. Holmes further argues that set theory is more natural with than without urelements, since we may take as urelements the objects of any theory or of the physical universe.[6]