Equivalence of Definitions of Strict Well-Ordering

Proof
Let $A$ be a class.

Let $B$ be a non-empty subset of $A$.

Let $\prec$ be a relation on $A$.

Definition 1 implies Definition 2
Suppose that $\prec$ is a strictly well-founded strict total ordering of $A$.

By the definition of strict total ordering, $\prec$ connects $A$.

Definition 2 implies Definition 1
Suppose that $\prec$ connects $A$ and that $\prec$ is a strictly well-founded relation.

That is:
 * for any $x, y \in A$, either $x = y$, $x \prec y$, or $y \prec x$
 * whenever $b$ is a non-empty subset of $A$, $b$ has a $\prec$-minimal element.

$\prec$ is transitive:

Let $x, y, z \in A$.

Let $x \prec y$ and $y \prec z$.

Since $\prec$ connects $A$, either $x \prec z$ or $z \prec x$.

By Strictly Well-Founded Relation has no Relational Loops it is not the case that $x \prec y$, $y \prec z$ and $z \prec x$.

Thus we conclude that $x \prec z$.

As this holds for all such $x, y, z$, $\prec$ is transitive.

$\prec$ is antireflexive by Strictly Well-Founded Relation is Antireflexive.

Thus $\prec$ is a strict total ordering.