Theorem ltslin

Description: Surreal less-than obeys trichotomy. (Contributed by Scott Fenton, 16-Jun-2011)

	Ref	Expression
Assertion	ltslin	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐴 <s 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 <s 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		ltsso	⊢ <s Or No
2		solin	⊢ ( ( <s Or No ∧ ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ) → ( 𝐴 <s 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 <s 𝐴 ) )
3	1 2	mpan	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐴 <s 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 <s 𝐴 ) )