Well, if your logical system defines "not P" as "assuming P, you can derive falsity", then "uncountable" precisely means "if you assume it is countable then you

Well, if your logical system defines "not P" as "assuming P, you can derive falsity", then "uncountable" precisely means "if you assume it is countable then you can derive falsity". That's exactly what the proof did. If you particularly care then you can just replace the final word "contradiction" with "QED"; it doesn't change the structure or content of the proof.