Finite Ordinal Plus Transfinite Ordinal

Theorem
Let $n$ be a finite ordinal.

Let $x$ be a transfinite ordinal.

Then:
 * $n + x = x$

Proof
By Transfinite Induction on $x$.

The proof will use $<$, $\in$, and $\subset$ interchangeably. This is justified by Transitive Set is Proper Subset of Ordinal iff Element of Ordinal.

Base Case
By our hypothesis, $\omega \le x$, so $x \nless \omega$, so we may begin our induction at $\omega$.

From these conclusions, we may deduce that:
 * $\ds \omega = \bigcup_{y \mathop \in \omega} \paren {n + y}$