No Natural Number between Number and Successor/Proof using Minimally Inductive Set

Proof
We use the model that $\N \cong \omega$ where $\omega$ is the minimally inductive set.

such an ordinal $m$ exists.

Then, by Ordering on Ordinal is Subset Relation:


 * $n \in m$

and from Transitive Set is Proper Subset of Ordinal iff Element of Ordinal:


 * $m \in n^+$

Applying the definition of a successor set, we have:


 * $n \in m \lor n = m$

But this creates a membership loop, in that:


 * $m \in n \in m \lor m \in m$

By No Membership Loops, we have created a contradiction.

The result follows from Proof by Contradiction.