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	$$⊢A\in {\mathbb{N}}_{\mathrm{0s}}\leftrightarrow \left(A\in {On}_s\wedge \mathrm{bday}\left(A\right)\in \mathrm{\omega }\right)$$

Proof

Step	Hyp	Ref	Expression
1		n0on	$$⊢A\in {\mathbb{N}}_{\mathrm{0s}}\to A\in {On}_s$$
2		n0bday	$$⊢A\in {\mathbb{N}}_{\mathrm{0s}}\to \mathrm{bday}\left(A\right)\in \mathrm{\omega }$$
3	1 2	jca	$$⊢A\in {\mathbb{N}}_{\mathrm{0s}}\to \left(A\in {On}_s\wedge \mathrm{bday}\left(A\right)\in \mathrm{\omega }\right)$$
4		onsfi	$$⊢\left(A\in {On}_s\wedge \mathrm{bday}\left(A\right)\in \mathrm{\omega }\right)\to A\in {\mathbb{N}}_{\mathrm{0s}}$$
5	3 4	impbii	$$⊢A\in {\mathbb{N}}_{\mathrm{0s}}\leftrightarrow \left(A\in {On}_s\wedge \mathrm{bday}\left(A\right)\in \mathrm{\omega }\right)$$