Equivalence of Definitions of Strict Well-Ordering

Theorem
The following definitions of strict well-ordering are equivalent:

Proof
Let $A$ be a class.

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

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

Definition 1 implies Definition 2
Suppose that $\prec$ is a foundational strict total ordering of $A$.

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

Let $B$ be a subclass of $A$.

By Well-Founded Relation Determines Minimal Elements, $\prec$ is strongly well-founded.

Thus $B$ has an $\prec$-minimal element, $x$.

Definition 2 implies Definition 1
Suppose that $\prec$ connects $A$ and that $\prec$ is a strongly 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 subclass of $A$, $B$ has a $\prec$-minimal element.

Since $\prec$ is strongly well-founded, it is trivially a foundational relation.

$\prec$ 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 Foundational 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 Foundational Relation is Antireflexive.

Thus $\prec$ is a strict total ordering.