Theorem ltnles

Description: Surreal less-than in terms of less-than or equal. (Contributed by Scott Fenton, 8-Dec-2021)

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

Proof

Step	Hyp	Ref	Expression
1		lenlts	⊢ ( ( 𝐵 ∈ No ∧ 𝐴 ∈ No ) → ( 𝐵 ≤s 𝐴 ↔ ¬ 𝐴 <s 𝐵 ) )
2	1	ancoms	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐵 ≤s 𝐴 ↔ ¬ 𝐴 <s 𝐵 ) )
3	2	con2bid	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐴 <s 𝐵 ↔ ¬ 𝐵 ≤s 𝐴 ) )