Limit Ordinal Equals its Union

Theorem
Let $\lambda$ be a limit ordinal.

Then:
 * $\lambda = \bigcup \lambda$

where $\bigcup \lambda$ denotes the union of $\lambda$.

Proof
From Union of Ordinal is Subset of Itself:
 * $\bigcup \lambda \subseteq \lambda$

Suppose $x \in \lambda$.

By Successor of Ordinal Smaller than Limit Ordinal is also Smaller:
 * $x^+ < \lambda$

and so:
 * $x \in x^+$ and $x^+ \in \lambda$

from which:
 * $x \in \bigcup \lambda$

That is:
 * $\lambda \subseteq \bigcup \lambda$

Hence by set equality:
 * $\lambda = \bigcup \lambda$

Also presented as
This can also be presented in the form:
 * $\lambda = \ds \bigcup_{\alpha \mathop \in \lambda} \alpha$

which by definition of union can be seen to be equivalent to $\lambda = \bigcup \lambda$.