Do you also see the First Incompleteness Theorem, or Halting Problem, as sleights of hand? Informally, the answer to your question is no - the Schröder-Bernste

Do you also see the First Incompleteness Theorem, or Halting Problem, as sleights of hand? Informally, the answer to your question is no - the Schröder-Bernstein theorem, which lets us order the size of sets, is sufficient to derive the law of the excluded middle. Therefore if you don't like the "trick" i.e. proof by contradiction (even given the contradiction is "actually constructed" in this case), and instead demand constructive mathematics, you will not be able to say much about relative cardinality.