Theorem ltstr

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

	Ref	Expression
Assertion	ltstr	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( ( 𝐴 <s 𝐵 ∧ 𝐵 <s 𝐶 ) → 𝐴 <s 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		ltsso	⊢ <s Or No
2		sotr	⊢ ( ( <s Or No ∧ ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ) → ( ( 𝐴 <s 𝐵 ∧ 𝐵 <s 𝐶 ) → 𝐴 <s 𝐶 ) )
3	1 2	mpan	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( ( 𝐴 <s 𝐵 ∧ 𝐵 <s 𝐶 ) → 𝐴 <s 𝐶 ) )