Ordinal Membership is Trichotomy/Corollary

Corollary to Ordinal Membership is Trichotomy

Let $\alpha$ be an ordinal.

Let $x, y \in \alpha$ such that $x \ne y$.

Then either:

$x \in y$

or:

$y \in x$

Proof

We have that Element of Ordinal is Ordinal.

The result then follows directly from Ordinal Membership is Trichotomy.

$\blacksquare$

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $5$: Ordinal Numbers: $\S 1$ Ordinal numbers: Corollary $1.15$