Def/Ordinal number

• If $x,y \in S$, then $x$ and $y$ are comparable, since $S$ is well-ordered and hence totally ordered.
• If $x \in S$, and $y = S$, then $x \in y$, so $x < y$.
• If $x = S$, and $y \in S$, then $y \in x$, so $y < x$.
• If $x = S$ and $y = S$, then $x = y$.

Thus, $x$ is always comparable to $y$.

• If $M \subset S$, then $M$ has a least element, since $S$ is well-ordered.
• If $M = \{ S \}$, then $S$ is the unique, and hence least, element of $M$.
• Otherwise, $M = N \cup \{ S \}$, where $N$ is a subset of $S$. Since $S$ is greater than all of its elements (by the axiom of regularity), it follows that the least element of $N$ (which exists since $S$ is well-ordered) is also the least element of $M$.