Theorem exeupre2

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

	Ref	Expression
Assertion	exeupre2	⊢ ( ∃ 𝑚 suc 𝑚 = 𝑁 ↔ ∃! 𝑚 suc 𝑚 = 𝑁 )

Proof

Step	Hyp	Ref	Expression
1		mopre	⊢ ∃* 𝑚 suc 𝑚 = 𝑁
2		moeuex	⊢ ( ∃* 𝑚 suc 𝑚 = 𝑁 → ( ∃ 𝑚 suc 𝑚 = 𝑁 ↔ ∃! 𝑚 suc 𝑚 = 𝑁 ) )
3	1 2	ax-mp	⊢ ( ∃ 𝑚 suc 𝑚 = 𝑁 ↔ ∃! 𝑚 suc 𝑚 = 𝑁 )