Category:Union of Initial Segments is Initial Segment or All of Woset

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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 $X$.