My math prof used this result in a proof about sequences without justification. I tried to prove it myself (as an exercise), but my proof quickly got out of control. I ended up using set cardinality and the injectivity of strictly increasing functions. My (unfinished) proof also depends on the truth value of [|A| > |B|] ⇒ A ⊈ B, which I think will be even harder to prove (if it is not axiomatic). Is there a more obvious way to prove this result? I feel like there must be because my prof stated it without explaining.