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ZFC set theory is said to be FOL with one non-logical symbol $\in$(ref). However, I have not seen a book/material about ZFC that only uses $\in$. For example, in https://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html, apart from $\in$, at least $\cap$, $\cup$,{,},$\emptyset$ are used. I wonder what roles these symbols play in FOL. Is $\emptyset$ a constant symbol of FOL? How to write the ZFC axioms in pure FOL(that includes the only non-logical symbol $\in$)?

William
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    You might enjoy a programming course where one writes the program in pure machine language, i.e. 0's and 1's. – Lee Mosher Oct 14 '24 at 13:44

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Using $\cup$ is merely a shorthand. Instead of saying that $x\in A\cup B$ one could say $x\in A \vee x\in B$. The existence of the empty set can either be postulated as a separate axiom, or derived from the axiom of infinity (which does seem like a roundabout way of doing it, but is perhaps more economical).

Mikhail Katz
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