Theorem eln0s2

Description: A non-negative surreal integer is a surreal ordinal with a finite birthday. (Contributed by Scott Fenton, 27-Feb-2026)

	Ref	Expression
Assertion	eln0s2	⊢ ( 𝐴 ∈ ℕ0s ↔ ( 𝐴 ∈ Ons ∧ ( bday ‘ 𝐴 ) ∈ ω ) )

Proof

Step	Hyp	Ref	Expression
1		n0on	⊢ ( 𝐴 ∈ ℕ0s → 𝐴 ∈ Ons )
2		n0bday	⊢ ( 𝐴 ∈ ℕ0s → ( bday ‘ 𝐴 ) ∈ ω )
3	1 2	jca	⊢ ( 𝐴 ∈ ℕ0s → ( 𝐴 ∈ Ons ∧ ( bday ‘ 𝐴 ) ∈ ω ) )
4		onsfi	⊢ ( ( 𝐴 ∈ Ons ∧ ( bday ‘ 𝐴 ) ∈ ω ) → 𝐴 ∈ ℕ0s )
5	3 4	impbii	⊢ ( 𝐴 ∈ ℕ0s ↔ ( 𝐴 ∈ Ons ∧ ( bday ‘ 𝐴 ) ∈ ω ) )