Theorem dfpre3

Description: Alternate definition of the successor-predecessor. (Contributed by Peter Mazsa, 12-Jan-2026)

	Ref	Expression
Assertion	dfpre3	⊢ ( 𝑁 ∈ 𝑉 → pre 𝑁 = ( ℩ 𝑚 suc 𝑚 = 𝑁 ) )

Proof

Step	Hyp	Ref	Expression
1		dfpre2	⊢ ( 𝑁 ∈ 𝑉 → pre 𝑁 = ( ℩ 𝑚 𝑚 SucMap 𝑁 ) )
2		brsucmap	⊢ ( ( 𝑚 ∈ V ∧ 𝑁 ∈ 𝑉 ) → ( 𝑚 SucMap 𝑁 ↔ suc 𝑚 = 𝑁 ) )
3	2	el2v1	⊢ ( 𝑁 ∈ 𝑉 → ( 𝑚 SucMap 𝑁 ↔ suc 𝑚 = 𝑁 ) )
4	3	iotabidv	⊢ ( 𝑁 ∈ 𝑉 → ( ℩ 𝑚 𝑚 SucMap 𝑁 ) = ( ℩ 𝑚 suc 𝑚 = 𝑁 ) )
5	1 4	eqtrd	⊢ ( 𝑁 ∈ 𝑉 → pre 𝑁 = ( ℩ 𝑚 suc 𝑚 = 𝑁 ) )