Is "$\subset$" a symbol of first-order language of set theory ?

in Mathematical introduction to logic by Enderton , he says that the only 2-place predicate is $\in$ , but i can't understand why $\subset$ doesn't exist ? how can we deal with set theory without subset notion ?

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    $\begingroup$ Instead of saying that $A\subset X$ you can say $A\in P(X)$. Is this the reason? $\endgroup$ – Sigur Jun 25 '13 at 1:32
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    $\begingroup$ @Sigur: Not really, since the power set symbol isn’t in the language either. $\endgroup$ – Brian M. Scott Jun 25 '13 at 1:38
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    $\begingroup$ No, I mean that just as $\subseteq$ is not a symbol of the language, so also the $\wp$ of $\wp(x)$ is not a symbol of the language. $\endgroup$ – Brian M. Scott Jun 25 '13 at 1:40
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    $\begingroup$ @MathsLover Because $P(X)$ can be defined as the set of all subsets of $X$, where subset is defined as in B.Scott's answer below. The symbol is just shorthand. $\endgroup$ – Benjamin Dickman Jun 25 '13 at 1:40
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    $\begingroup$ There have been occasional attempts to make a set theory that is based on the subset relation, not the membership relation. In that case, $\subset$ would be a symbol of the language, and $\in$ would not be! None of the attempts has drawn serious attention from the mathematical community. $\endgroup$ – André Nicolas Jun 25 '13 at 1:41

You don’t need to have a two-place predicate $\subseteq$ in the language, because you can define it by a formula: $x\subseteq y$ is an abbreviation for

$$\forall z(z\in x\to z\in y)\;.$$

One ends up defining a great many things that aren’t actually in the formal language: $0$, $\omega$, $\wp(x)$, $\bigcup x$, etc., but keeping the formal language to a minimum simplifies some technical arguments quite a bit — anything that involves induction on formulas, for instance.

  • $\begingroup$ Oh! Yes, I remember now. Thanks. +1 $\endgroup$ – Sigur Jun 25 '13 at 1:40

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