基于特定有限/无限集定义的“非有限集必为无限集”命题验证问询

Hey everyone, I've got a question that might come off as trivial, but I'm actually stuck on it. It's rooted in set theory, so I'm operating under the assumptions that natural numbers are "God-given" (as in, we take them as a foundational given), and that we can use the Axiom of Choice along with all its equivalent forms—like the trichotomy of ordinals, Zorn's lemma, you name it—if needed. Just to clarify, I already know that every set is bijective to some ordinal.

First, let's formalize the definitions I'm working with:

• A set is finite if it is bijective to ${1,2,...,n}$ for some natural number $n \in \mathbb{N}$.
• A set $X$ is infinite if there exists a proper subset $A \subsetneq X$ that is bijective to $X$.

Here's the core question I'm grappling with:

Theorem (which might not actually hold): If a set is not finite, then it is infinite.

Put another way: If there is no bijection between a given set $X$ and ${1,2,...,n}$ for any $n \in \mathbb{N}$, can I prove that there must exist a proper subset $A \subsetneq X$ that is bijective to $X$?

备注：内容来源于stack exchange，提问作者Dibidus