Cantor-Bernstein-Schröder Theorem

Theorem
If a subset of one set is equivalent to the other, and a subset of the other is equivalent to the first, then the two sets are themselves equivalent:
 * $\forall S, T: T \sim S_1 \subseteq S \land S \sim T_1 \subseteq T \implies S \sim T$

Alternatively, from Dominates is Equivalent to Subset, this can be expressed as:
 * $\forall S, T: T \preccurlyeq S \land S \preccurlyeq T \implies S \sim T$

where $T \preccurlyeq S$ denotes the fact that $S$ dominates $T$.

That is:
 * If $\exists f: S \to T$ and $\exists g: T \to S$ where $f$ and $g$ are both injections, then there exists a bijection from $S$ to $T$.

Also known as

 * The Cantor-Bernstein Theorem
 * The Cantor-Schroeder-Bernstein Theorem or Cantor-Schröder-Bernstein Theorem
 * The Schroeder-Bernstein Theorem or Schröder-Bernstein Theorem

Comments

 * This theorem states in set theoretical concepts the "intuitively obvious" fact that if $a \le b$ and $b \le a$ then $a = b$.

Care needs to be taken to make well sure of this, because when considering infinite sets, intuition is frequently misleading.


 * In order to prove equivalence, a bijection needs to be demonstrated. It can be significantly simpler to demonstrate an injection than a surjection, so proving that there is an injection from $S$ to $T$ and also one from $T$ to $S$ may be a lot less work than proving that there is both an injection and a surjection from $S$ to $T$.

Cantor first attempted to prove this theorem in his 1897 paper. Ernst Schröder had also stated this theorem some time earlier, but his proof, as well as Cantor's, was flawed. It was Felix Bernstein who finally supplied a correct proof in his 1898 PhD thesis.