Theorem sltasym

Description: Surreal less-than is asymmetric. (Contributed by Scott Fenton, 16-Jun-2011)

	Ref	Expression
Assertion	sltasym	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐴 <s 𝐵 → ¬ 𝐵 <s 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		sltso	⊢ <s Or No
2		soasym	⊢ ( ( <s Or No ∧ ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ) → ( 𝐴 <s 𝐵 → ¬ 𝐵 <s 𝐴 ) )
3	1 2	mpan	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐴 <s 𝐵 → ¬ 𝐵 <s 𝐴 ) )