Ordinal is Well-Ordered by Epsilon

Theorem

If $A$ is an ordinal number, then the epsilon relation $\Epsilon$ is a strict well ordering on $A$.

Proof

From Epsilon is Foundational, $\Epsilon$ is a foundational relation on $A$.

By No Membership Loops, $\neg A \in A$.

By the fact that Every Ordinal is a Transitive Class:

$B \in A \implies B \subseteq A$

Therefore, it satisfies transitivity:

$(B \in A \land C \in B) \implies C \in A$

Finally, by Relation between Unequal Ordinals, if $x$ and $y$ are ordinals, then:

$x \subset y \lor x = y \lor y \subset x$

Since, by Initial Segment of Ordinal is Ordinal, all elements of $A$ are themselves ordinals, it follows that:

$\forall x, y \in A: (x \subset y \lor x = y \lor y \in x)$

Because $A$ is an ordinal:

$\forall x \in A: \{ y \in A : y \subset x \} = x$

Therefore, the $\subset$ relation is the same as membership between two ordinals, so:

$\forall x, y \in A: (x \in y \lor x = y \lor y \in x)$

The second and third lines show that the Ordinals are ordered by $\Epsilon$.

The fourth shows that this ordering is a strict total ordering.

Finally, the first shows that this ordering is a strict well-ordering.

$\blacksquare$

See Also

• Axiom of Foundation
• Foundational Relation