Theorem sltletr

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

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

Proof

Step	Hyp	Ref	Expression
1		slenlt	⊢ ( ( 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐵 ≤s 𝐶 ↔ ¬ 𝐶 <s 𝐵 ) )
2	1	3adant1	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐵 ≤s 𝐶 ↔ ¬ 𝐶 <s 𝐵 ) )
3	2	anbi2d	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( ( 𝐴 <s 𝐵 ∧ 𝐵 ≤s 𝐶 ) ↔ ( 𝐴 <s 𝐵 ∧ ¬ 𝐶 <s 𝐵 ) ) )
4		sltso	⊢ <s Or No
5		sotr3	⊢ ( ( <s Or No ∧ ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ) → ( ( 𝐴 <s 𝐵 ∧ ¬ 𝐶 <s 𝐵 ) → 𝐴 <s 𝐶 ) )
6	4 5	mpan	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( ( 𝐴 <s 𝐵 ∧ ¬ 𝐶 <s 𝐵 ) → 𝐴 <s 𝐶 ) )
7	3 6	sylbid	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( ( 𝐴 <s 𝐵 ∧ 𝐵 ≤s 𝐶 ) → 𝐴 <s 𝐶 ) )