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On Tent and Ziegler's textbook "Model Theory", it is stated that the language of Set Theory contains only the binary relation $\in$. How is that possible, as ZFC contains only sets and $\in$ is a subset of a cartesian product between a set and an urelement, as stated here What is the difference between the relations $\in$ and $\subseteq$??

Also, in the ZFC Axioms in Wolfram's Mathworld (http://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html), the Power Set Axiom string contains the $\subseteq$ relation, which according to Tent and Ziegler, is not part of the ZFC language.

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$\subseteq$ relation can be written with $\in $ relation as $$A \subseteq B \text{ is defined by } \forall x (x \in A \Rightarrow x \in B)$$

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  • $\begingroup$ For this to be valid, shouldn't it be listed among the ZFC Axioms? $\endgroup$ – Bruno Schiavo Aug 11 '15 at 14:11
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    $\begingroup$ No. $\subseteq$ is just a way to define a relation based on $\in$. It is just a way to simplify the writing and avoid to write $\forall x (x \in A \Rightarrow x \in B)$ each time you want to mean $A \subseteq B$. $\endgroup$ – mathcounterexamples.net Aug 11 '15 at 14:14
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A relation-as-part-of-formal-language isn't the same thing as a relation-as-type-of-set. In the former case it's a piece of syntax--e.g. the symbol '$\in$' along with the information that it takes two term symbols to make a formula out if it. In the latter case it is a subset of the Cartesian product.

Note that you can sniff out that these are distinct concepts by noting that they don't always match up; in most set theories there's no $X$ such that $\langle x,y\rangle\in X\Leftrightarrow x\in y$.

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