Assume that D ⊂ N and D ̸= ∅. Prove or disprove using a detailed structured proof, justifying every step: [∀x ∈ D, ∃y ∈ N, y < x] ⇔ [0 ̸∈ D]
I have no idea how to prove a statement like that, I'm completely stuck! If anyone could help me out in any way that would be awesome
Hint: This appears to be about the Well-Ordering Principle, which states that every nonempty subset $D$ of the natural numbers $\mathbb N$ has a least element (where I assume that you define the natural numbers to start at $0$ instead of at $1$). That is, we know that if $D \subseteq \mathbb N$ and $D \neq \emptyset$, then: $$ \exists y \in D \text{ such that } \forall x \in D,~ y \leq x $$
Hence, if $y$ is allowed to be any natural number and if we want that inequality to be strict, then we want to exclude the possibility that $x = 0 \in D$ so that we can just take $y = 0$ to be the natural number that is strictly smaller than any other natural number.
