Existence of Minimal Uncountable Well-Ordered Set/Corollary 2

Corollary to Existence of Minimal Uncountable Well-Ordered Set
Let $X$ be any well-ordered set.

Then there exists a well-ordered set with cardinality strictly greater than $X$.

Proof
Follows from the same arguments proving the main result, mutatis mutandis, replacing:


 * "$\mathcal P \left({\N}\right)$" with "$\mathcal P \left({X}\right)$"


 * "Power Set of Natural Numbers Not Countable" with "No Bijection from Set to its Power Set"


 * "is uncountable" with "has cardinality $\left \vert {\mathcal P \left({X}\right)}\right\vert$"


 * "is countable" with "has cardinality strictly less than $\left \vert {\mathcal P \left({X}\right)}\right\vert$"