Ordinals are Well-Ordered

Theorem
The ordinals are well-ordered.

Proof
Recall that the ordinals are totally ordered.

Let $A$ be a non-empty set of ordinals.

Let $\alpha \in A$.

Let $B = \alpha^{+} \cap A$, where $\alpha^{+}$ denotes the successor set of $\alpha$.

Recall that $\alpha^{+}$ is an ordinal.

Note that $\alpha \in B$, so $B$ is non-empty.

By Intersection Subset, $B \subseteq \alpha^{+}$; therefore, there exists a smallest element $\kappa$ of $B$.

We claim that $\kappa$ is the smallest element of $A$.

So let $\beta \in A$. We need to show that $\kappa = \beta$ or $\kappa \in \beta$.

By Ordinal Membership Trichotomy, either $\beta \in \alpha^{+}$, $\alpha^{+} = \beta$, or $\alpha^{+} \in \beta$.

If $\beta \in \alpha^{+}$, then $\beta \in B$; it follows by the definition of $\kappa$ that $\kappa = \beta$ or $\kappa \in \beta$.

If $\alpha^{+} = \beta$ or $\alpha^{+} \in \beta$, then it follows by the transitivity of $\beta$ that $\alpha^{+} \subseteq \beta$.

Since $\kappa \in \alpha^{+}$, it follows that $\kappa \in \beta$.