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

Sufficient Condition
Suppose that $x < y$.

By Subset is Right Compatible with Ordinal Exponentiation:
 * $x^w \le y^w$

Then:

Thus the sufficient condition is satisfied.

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:
 * $y \not \le x$

By Ordinal Membership is Trichotomy:
 * $x < y$