Union of Set of Ordinals is Ordinal/Proof 2

Theorem

Let $A$ be a set of ordinals.

Then $\bigcup A$ is an ordinal.

Proof

$\ds x$

$\in$

$\ds \bigcup A$

$\ds \leadsto \ \ $

$\ds \exists y \in A: \, $

$\ds x$

$\in$

$\ds y$

$\ds \leadsto \ \ $

$\ds \exists y \subseteq \bigcup A: \, $

$\ds x$

$\subseteq$

$\ds y$

$\ds \leadsto \ \ $

$\ds x$

$\subseteq$

$\ds \bigcup A$

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