Theorem onltn0s

Description: A surreal ordinal that is less than a non-negative integer is a non-negative integer. (Contributed by Scott Fenton, 7-Nov-2025)

	Ref	Expression
Assertion	onltn0s	⊢ ( ( 𝐴 ∈ Ons ∧ 𝐵 ∈ ℕ0s ∧ 𝐴 <s 𝐵 ) → 𝐴 ∈ ℕ0s )

Proof

Step	Hyp	Ref	Expression
1		simp1	⊢ ( ( 𝐴 ∈ Ons ∧ 𝐵 ∈ ℕ0s ∧ 𝐴 <s 𝐵 ) → 𝐴 ∈ Ons )
2		n0ons	⊢ ( 𝐵 ∈ ℕ0s → 𝐵 ∈ Ons )
3		onslt	⊢ ( ( 𝐴 ∈ Ons ∧ 𝐵 ∈ Ons ) → ( 𝐴 <s 𝐵 ↔ ( bday ‘ 𝐴 ) ∈ ( bday ‘ 𝐵 ) ) )
4	2 3	sylan2	⊢ ( ( 𝐴 ∈ Ons ∧ 𝐵 ∈ ℕ0s ) → ( 𝐴 <s 𝐵 ↔ ( bday ‘ 𝐴 ) ∈ ( bday ‘ 𝐵 ) ) )
5	4	biimp3a	⊢ ( ( 𝐴 ∈ Ons ∧ 𝐵 ∈ ℕ0s ∧ 𝐴 <s 𝐵 ) → ( bday ‘ 𝐴 ) ∈ ( bday ‘ 𝐵 ) )
6		n0sbday	⊢ ( 𝐵 ∈ ℕ0s → ( bday ‘ 𝐵 ) ∈ ω )
7	6	3ad2ant2	⊢ ( ( 𝐴 ∈ Ons ∧ 𝐵 ∈ ℕ0s ∧ 𝐴 <s 𝐵 ) → ( bday ‘ 𝐵 ) ∈ ω )
8		elnn	⊢ ( ( ( bday ‘ 𝐴 ) ∈ ( bday ‘ 𝐵 ) ∧ ( bday ‘ 𝐵 ) ∈ ω ) → ( bday ‘ 𝐴 ) ∈ ω )
9	5 7 8	syl2anc	⊢ ( ( 𝐴 ∈ Ons ∧ 𝐵 ∈ ℕ0s ∧ 𝐴 <s 𝐵 ) → ( bday ‘ 𝐴 ) ∈ ω )
10		onsfi	⊢ ( ( 𝐴 ∈ Ons ∧ ( bday ‘ 𝐴 ) ∈ ω ) → 𝐴 ∈ ℕ0s )
11	1 9 10	syl2anc	⊢ ( ( 𝐴 ∈ Ons ∧ 𝐵 ∈ ℕ0s ∧ 𝐴 <s 𝐵 ) → 𝐴 ∈ ℕ0s )