Zero is least element of ordinal

Question

Definition. An ordinal is a well-ordered set $X$ such that for all $x\in X$, $(−∞, x) = x$.

Lemma. Zero is least element of ordinal.

Proof. Let $\alpha$ be an ordinal. Let $x$ be least element of $\alpha$. So, $x=x\cap\alpha=\emptyset$. Thus $\emptyset$ is least element of $\alpha$, that is $0$ is least element of $\alpha$.

My questions: Why $x\cap\alpha=\emptyset$?

Answer 1

If $\alpha$ is an ordinal and $x\in\alpha$ then, according to the definition you just gave, $x=(-\infty,x)$ and so $x\cap\alpha=(-\infty,x)\cap\alpha.$ If $x$ is the least element of $\alpha,$ then $(-\infty,x) \cap\alpha=\emptyset.$