If $X$ and $Y$ are ordinals and $Y\subset X$, then $Y$ is a section of $X$

Question

I'm studying ordinals via Cameron's Sets, Logic and Categories. On p. 42, Cameron works to prove that

If $X$ and $Y$ are ordinals and $Y\subset X$, then $Y$ is a section of $X$.

Proof: Take $a$ to be the least element of $X\setminus Y$. Then $X_a\subseteq Y$. [The proof follows by showing that $Y\subseteq X_a$ by applying the trichotomy rule to $a$.]

I have two questions about this proof.

1. How does $X_a\subseteq Y$ follow immediately from $a$ being the least element of $X\setminus Y$? It isn't intuitive to me.
2. Wouldn't the proof proceed more simply as follows? Let $a$ be the least element of $X\setminus Y$. So $\forall y\in Y$, $y<a$ since $Y$ is an ordinal. Now $\forall x\in X$, if $x>a$ or $x=a$, $x\notin Y$. So $Y=\{x\in X:x<a\}=X_a$

Answer 1

By our choice of $a$ as the least element of $X\setminus Y,$ we have for any $b\in X_a$ that $b\in X=(X\cap Y)\cup(X\setminus Y)$ and $b<a,$ so $b\notin X\setminus Y,$ and so $b\in X\cap Y=Y.$ Thus $X_a\subseteq Y.$

Your approach starts off fine, applying the trichotomy rule to $a,$ and concluding that $Y\subseteq X_a.$ However, it doesn't show that $X_a\subseteq Y,$ only that $Y\subseteq X_a$ in another way (by contrapositive).