Prove that the set of irrational numbers in [0,1] is not countable

It is known that the set of rational numbers are countable.
Union of rational numbers and irrational numbers gives the real numbers.
So ${\displaystyle [0,1]=Q[0,1]\cup RQ[0,1]}$
Since Q[0,1] is countable and [0,1] is not countable.
So the only possibility is that ${\displaystyle RQ[0,1]}$ is uncountable.
Hence, the set of irrational numbers in [0,1] is not countable.