Surjection from Natural Numbers iff Countable/Corollary 2
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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.
$\blacksquare$