Condition for Membership is Right Compatible with Ordinal Exponentiation

Theorem

Let $x, y, z$ be ordinals.

Let $z$ be the successor of some ordinal $w$.

Then:

$x < y \iff x^z < y^z$

Proof

Sufficient Condition

Suppose $x < y$.

By Subset is Right Compatible with Ordinal Exponentiation:

$x^w \le y^w$

Then:

\(\ds x^z\)

\(=\)

\(\ds x^w \times x\)

Definition of Ordinal Exponentiation

\(\ds y^z\)

\(=\)

\(\ds y^w \times y\)

Definition of Ordinal Exponentiation

\(\ds x^w \times x\)

\(\le\)

\(\ds y^w \times x\)

Subset is Right Compatible with Ordinal Exponentiation

\(\ds \)

\(<\)

\(\ds y^w \times y\)

Membership is Left Compatible with Ordinal Exponentiation

Thus the sufficient condition is satisfied.

$\Box$

Necessary Condition

Suppose $x^z < y^z$.

From Subset is Right Compatible with Ordinal Exponentiation:

$y \le x \implies y^z \le x^z$

This contradicts:

$x^z < y^z$

so:

$y \nleq x$

By Ordinal Membership is Trichotomy:

$x < y$

$\blacksquare$

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 8.36$