Set which is Equivalent to Countable Set is Countable

Lemma

Let $S$ be countable set.

Let $T$ be a set.

Let $T$ be equivalent to $S$.

Then $T$ is countable.

Proof

By definition of set equivalence:

$S$ is in one-to-one correspondence with $T$.

We have that $S$ is countable.

By definition of countable set:

$S$ is in one-to-one correspondence with a subset of the natural numbers.

$T$ is in one-to-one correspondence with $S$.

By Composite of Bijections is Bijection:

$T$ is in one-to-one correspondence with a subset of the natural numbers.

Hence by definition of countable set: $T$ is countable.

$\blacksquare$