Comparability of cardinals

Statement

Assume the axiom of choice. Let $A$ and $B$ be sets. Then there is an injective function from $A$ to $B$ or an injective function from $B$ to $A$. So $|A| \le |B|$ or $|B| \le |A|$.

Remarks

With the Schröder–Bernstein theorem, it follows that the order of cardinals is a total order. The statement is equivalent to the axiom of choice over ZF. One proof uses Zorn's lemma on the set of injective partial functions from $A$ to $B$.

Explanations

L1 human

Any two sets can be compared in size: one of them fits into the other. Without the axiom of choice, two sets can exist such that neither fits into the other.