User:Dfeuer/Successor Mapping is Injective
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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 $m = n$.
$\blacksquare$