True, though this "proof" assumes a large amount of additional mathematical machinery in addition to SB (I learned it as Cantor-Schröder–Bernstein originally...

True, though this "proof" assumes a large amount of additional mathematical machinery in addition to SB (I learned it as Cantor-Schröder–Bernstein originally... I wonder where the inconsistency in the naming comes from). It's not so much a formal set-theoretic proof (of course it could be made so) as a quick demonstration that has a stronger pull on mathematical intuition than the standard proof (and is demonstrating an otherwise provable thing).