User:Dfeuer/Successor Mapping is Injective

Theorem
Let $m$ and $n$ be natural numbers.

Suppose that $m^+ = n^+$.

Then $m = n$.

Proof
$m^+ = m \cup \{m\}$ and $n^+ = n \cup \{n\}$.

Thus by the definitions of singleton and union:


 * $m \in m \cup \{m\}$ and $n \in n \cup \{n\} $.

Thus $m \in n \cup \{n\}$ and $n \in m \cup \{m\}$.

Thus:


 * $m \in n$ or $m = n$
 * $n \in m$ or $m = n$

So $m = n \lor (m \in n \land n \in m)$.

By Membership is Asymmetric on Natural Numbers, $\lnot (m \in n \land n \in m)$.

Thus $m = n$.