Theorem ltsnled

Description: Surreal less-than in terms of less-than or equal. Deduction version. (Contributed by Scott Fenton, 25-Feb-2026)

	Ref	Expression
Hypotheses	lesd.1	⊢ ( 𝜑 → 𝐴 ∈ No )
	lesd.2	⊢ ( 𝜑 → 𝐵 ∈ No )
Assertion	ltsnled	⊢ ( 𝜑 → ( 𝐴 <s 𝐵 ↔ ¬ 𝐵 ≤s 𝐴 ) )

Proof

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