Theorem exeupre

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

	Ref	Expression
Assertion	exeupre	⊢ ( 𝑁 ∈ 𝑉 → ( ∃ 𝑚 𝑚 SucMap 𝑁 ↔ ∃! 𝑚 𝑚 SucMap 𝑁 ) )

Proof

Step	Hyp	Ref	Expression
1		brsucmap	⊢ ( ( 𝑚 ∈ V ∧ 𝑁 ∈ 𝑉 ) → ( 𝑚 SucMap 𝑁 ↔ suc 𝑚 = 𝑁 ) )
2	1	el2v1	⊢ ( 𝑁 ∈ 𝑉 → ( 𝑚 SucMap 𝑁 ↔ suc 𝑚 = 𝑁 ) )
3	2	exbidv	⊢ ( 𝑁 ∈ 𝑉 → ( ∃ 𝑚 𝑚 SucMap 𝑁 ↔ ∃ 𝑚 suc 𝑚 = 𝑁 ) )
4		exeupre2	⊢ ( ∃ 𝑚 suc 𝑚 = 𝑁 ↔ ∃! 𝑚 suc 𝑚 = 𝑁 )
5	3 4	bitrdi	⊢ ( 𝑁 ∈ 𝑉 → ( ∃ 𝑚 𝑚 SucMap 𝑁 ↔ ∃! 𝑚 suc 𝑚 = 𝑁 ) )
6	2	eubidv	⊢ ( 𝑁 ∈ 𝑉 → ( ∃! 𝑚 𝑚 SucMap 𝑁 ↔ ∃! 𝑚 suc 𝑚 = 𝑁 ) )
7	5 6	bitr4d	⊢ ( 𝑁 ∈ 𝑉 → ( ∃ 𝑚 𝑚 SucMap 𝑁 ↔ ∃! 𝑚 𝑚 SucMap 𝑁 ) )