Theorem exeupre2

Description: Whenever a predecessor exists, it exists alone. (Contributed by Peter Mazsa, 12-Jan-2026)

	Ref	Expression
Assertion	exeupre2	$$⊢\exists m\mathrm{suc}m=N\leftrightarrow \exists !m\mathrm{suc}m=N$$

Proof

Step	Hyp	Ref	Expression
1		mopre	$$⊢{\exists }^{*}m\mathrm{suc}m=N$$
2		moeuex	$$⊢{\exists }^{*}m\mathrm{suc}m=N\to \left(\exists m\mathrm{suc}m=N\leftrightarrow \exists !m\mathrm{suc}m=N\right)$$
3	1 2	ax-mp	$$⊢\exists m\mathrm{suc}m=N\leftrightarrow \exists !m\mathrm{suc}m=N$$