Let $X$ be a set. I need to find an injection from the set of ordinals $$ \Gamma(x) = \{ \alpha : \text{there exists an injection $f$ such that} \quad f:\alpha \to X\}$$
to $\mathcal{P}(\mathcal{P}(X \times X))$ (without using $AC$).
My reasoning: Since for $\alpha \in \Gamma(X)$ there exists an injection $f : \alpha \to X$, I am thinking on using $f[\alpha] \subseteq X$ and somehow build an injection to $\mathcal{P}(\mathcal{P}(X \times X))$ but I am lost. Any hint would be appreciated!
For $\alpha\in\Gamma(X) ,$ let $g(\alpha)$ be the set of all well-orderings of $X$ of type $\alpha.$ This function $g$ is injective, and any well-ordering is a subset of $X\times X,$ so an element of $\mathcal P(X\times X),$ so a set of well-orderings of $X$ is an element of $\mathcal P(\mathcal P(X\times X)).$