Hartogs' Lemma (Set Theory)

Theorem
Let $X$ be a set. There is an ordinal $\alpha$ such that there is no injection from $\alpha$ to $X$.

(Note: This does not depend on the axiom of choice)

Proof
Let $\alpha = \{ \beta : \beta $ is an ordinal and there is an injection from $\beta \to X \}$

This is a well defined set, because it is the image of all well orderings of subsets of $X$ under the order-type mapping.

It is an ordinal because if $\beta \in \alpha$ and $\gamma < \beta$ because if $i : \beta \to X$ is an injection then to $i|_\gamma$ is an injection $\gamma \to X$.

There can be no injection $\alpha \to X$ because if there were then we would have $\alpha \in \alpha$, which contradicts the well foundedness of ordinals.