Theorem ltstrine

Description: Trichotomy law for surreals. (Contributed by Scott Fenton, 23-Nov-2021)

	Ref	Expression
Assertion	ltstrine	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐴 ≠ 𝐵 ↔ ( 𝐴 <s 𝐵 ∨ 𝐵 <s 𝐴 ) ) )

Proof

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