Schröder–Bernstein theorem

Statement

Let $A$ and $B$ be sets. If there are injective functions $f \colon A \to B$ and $g \colon B \to A$, then there is a bijective function $h \colon A \to B$.

Remarks

In terms of cardinals: $|A| \le |B|$ and $|B| \le |A|$ imply $|A| = |B|$. So the order of cardinals is antisymmetric. The proof does not use the axiom of choice. Example: $[0, 1]$ and $[0, 1)$ are equinumerous, because each one injects into the other.

History

Cantor stated the theorem in 1887 without a proof. Dedekind proved it in 1887, but he did not publish the proof. Bernstein and Schröder published proofs in 1898, and the proof of Schröder had a flaw.

Explanations

L1 human

If $A$ fits into $B$ and $B$ fits into $A$, then $A$ and $B$ have the same size. For finite sets this is obvious, but for infinite sets it needs a clever construction of $h$.