Non-Finite Cardinal is equal to Cardinal Product

Theorem
Let $\omega$ denote the minimal infinite successor set.

Let $x$ be an ordinal such that $x \ge \omega$.

Then:


 * $\left|{ x }\right| = \left|{ x \times x }\right|$

Where $\times$ denotes the Cartesian product.

Proof
The proof shall proceed by Transfinite Induction on $x$.