Theorem sletri3d

Description: Trichotomy law for surreal less-than or equal. Deduction version. (Contributed by Scott Fenton, 25-Feb-2026)

	Ref	Expression
Hypotheses	sled.1	⊢ ( 𝜑 → 𝐴 ∈ No )
	sled.2	⊢ ( 𝜑 → 𝐵 ∈ No )
Assertion	sletri3d	⊢ ( 𝜑 → ( 𝐴 = 𝐵 ↔ ( 𝐴 ≤s 𝐵 ∧ 𝐵 ≤s 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		sled.1	⊢ ( 𝜑 → 𝐴 ∈ No )
2		sled.2	⊢ ( 𝜑 → 𝐵 ∈ No )
3		sletri3	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐴 = 𝐵 ↔ ( 𝐴 ≤s 𝐵 ∧ 𝐵 ≤s 𝐴 ) ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝐴 = 𝐵 ↔ ( 𝐴 ≤s 𝐵 ∧ 𝐵 ≤s 𝐴 ) ) )