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 ?
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.