Hartogs bound on partial ordinal injections ($P_X\subseteq \mathcal{P}(h(X)\times X)$)

Every injection from an ordinal into a set $X$ has domain strictly below the Hartogs number $h(X)$. Therefore all partial injections with ordinal domain and range in $X$ form a set of graphs contained in $\mathcal{P}(h(X)\times X)$. Under extension, the union of a chain remains a partial injection of this kind. Zorn's lemma then supplies a maximal one, and extending at an unused range element shows that it must be a bijection onto $X$. This gives the well-ordering theorem with the set-sizedness of the poset made explicit.