Union of Initial Segments is Initial Segment or All of Woset

Theorem
Let $\struct {X, \preccurlyeq}$ be a well-ordered non-empty set.

Let $A \subseteq X$.

Let:


 * $\ds J = \bigcup_{x \mathop \in A} S_x$

be a union of initial segments defined by the elements of $A$.

Then either:


 * $J = X$

or:


 * $J$ is an initial segment of $X$.