Ordinal is Less than Ordinal times Limit

Theorem

Let $y$ be a limit ordinal.

Let $x$ and $z$ be ordinals.

Then:

$z < x \times y \iff \exists w \in y: z < x \times w$

Proof

\(\ds z\)

\(<\)

\(\ds x \times y\)

\(\ds \leadstoandfrom \ \ \)

\(\ds z\)

\(<\)

\(\ds \bigcup_{w \mathop \in y} \paren {x \times w}\)

Definition of Ordinal Multiplication

\(\ds \leadstoandfrom \ \ \)

\(\ds \exists w \in y: \, \)

\(\ds z\)

\(<\)

\(\ds x \times w\)

Definition of Set Union

$\blacksquare$

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 8.24$