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 Equivalence of Definitions of Dominate (Set Theory), 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 $T$.

Proof Strategy
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 $T$.

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