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:
 * $$\forall S, T: T \sim S_1 \subseteq S \and S \sim T_1 \subseteq T \implies S \sim T$$

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

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

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 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 $$T$$.