Theorem sltnled

Description: Surreal less-than in terms of 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	sltnled	⊢ ( 𝜑 → ( 𝐴 <s 𝐵 ↔ ¬ 𝐵 ≤s 𝐴 ) )

Proof

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