Understanding of definition of ordinal

Question

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

How could the definition say $(−∞, x) = x$, how can we obtain this? I didn't understand. Can you explain?

Answer 1

What this is saying is that each $x$ is precisely the set of all its predecessors in $X.$ In other words, letting $<$ be the order relation on $X$, for all $x\in X,$ we have $$x=\{t\in X:t<x\}.$$

The set-builder style set above is what the notation $(-\infty,x)$ means.

Answer 2

That notation means x is the set of all sets less than x (in the well-order). There is no actual -infinity, it is just used as part of the interval notation colorfully.