Theorem ltstrieq2

Description: Trichotomy law for surreal less-than. (Contributed by Scott Fenton, 22-Apr-2012)

	Ref	Expression
Assertion	ltstrieq2	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐴 = 𝐵 ↔ ( ¬ 𝐴 <s 𝐵 ∧ ¬ 𝐵 <s 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ltsso	⊢ <s Or No
2		sotrieq2	⊢ ( ( <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 ) ) )