Surjection from Natural Numbers iff Countable/Corollary 2

Theorem
Let $T$ be a countably infinite set.

Let $S$ be an uncountable set.

Let $f:T \to S$ be a mapping.

Then $f$ is not a surjection.

Proof
By Corollary 1 no mapping from $T$ to $S$ is a surjection.