Denumerable Class is Set

Theorem

Let $A$ be a class.

Let $\N$ be the natural numbers.

Suppose that $F: \N \to A$ is a bijection.

Then $A$ is a set.

Proof

By the Axiom of Infinity, $\N$ is a set.

Thus by the Axiom of Replacement, $A$ is also a set.

$\blacksquare$