Existence of Minimal Uncountable Well-Ordered Set/Corollary 2

Corollary to Existence of Minimal Uncountable Well-Ordered Set
Let $X$ be a 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:


 * "$\powerset \N$" with "$\powerset X$"


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


 * "is uncountable" with "has cardinality $\size {\powerset X}$"


 * "is countable" with "has cardinality strictly less than $\size {\powerset X}$"