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Proposition 187

[26J]This model of $ℕ$ is a well-ordered set with the ordering

n \leq m ⟺ n \subseteq m .

Moreover in this model we have

\begin{matrix} (1) & \forall n, m \in ℕ, n \in m ⟺ (n \subseteq m \land n \neq m) . \end{matrix}

188

so, defining (as usual)

n < m ≐ (n \leq m \land n \neq m)

we can write

n \in m ⟺ n < m .

This is proven in the following exercises, see in particular [269].

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Authors: "Mennucci , Andrea C. G." .

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