User:Dangcasanova/Sandbox.

Theorem
Let $x$ be a limit ordinal.

Then:


 * $\displaystyle x = \bigcup x$
 * $\displaystyle x = \bigcup_{y \mathop \in x} y$

Proof
Let $y\in x$. Then $y^+$ is an ordinal. Let us show that $y^+ \in x$.

By Theorem:Ordinal Membership is Trichotomy, $y^+ =x \or y^+ \in x \or x \in y^+$

Also see

 * Equivalence of Definitions of Oscillation at Point for Real Functions