Indirect Proofs of Existence and Non-existence

Theorem 9.1 (Cantor)   There is no onto function $A\to {\cal P}(A)$.

Theorem 9.2 (Euclid)   There are infinitely many prime numbers.

Theorem 9.3 (Cantor)   The set of real numbers is not countable.

Corollary 9.4   There are irrational numbers.

Corollary 9.5   There are transcendental numbers.