Theorem leltstr

Description: Surreal transitive law. (Contributed by Scott Fenton, 8-Dec-2021)

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

Proof

Step	Hyp	Ref	Expression
1		lenlts	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐴 ≤s 𝐵 ↔ ¬ 𝐵 <s 𝐴 ) )
2	1	3adant3	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐴 ≤s 𝐵 ↔ ¬ 𝐵 <s 𝐴 ) )
3	2	anbi1d	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( ( 𝐴 ≤s 𝐵 ∧ 𝐵 <s 𝐶 ) ↔ ( ¬ 𝐵 <s 𝐴 ∧ 𝐵 <s 𝐶 ) ) )
4		ltsso	⊢ <s Or No
5		sotr2	⊢ ( ( <s Or No ∧ ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ) → ( ( ¬ 𝐵 <s 𝐴 ∧ 𝐵 <s 𝐶 ) → 𝐴 <s 𝐶 ) )
6	4 5	mpan	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( ( ¬ 𝐵 <s 𝐴 ∧ 𝐵 <s 𝐶 ) → 𝐴 <s 𝐶 ) )
7	3 6	sylbid	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( ( 𝐴 ≤s 𝐵 ∧ 𝐵 <s 𝐶 ) → 𝐴 <s 𝐶 ) )