Metamath Proof Explorer


Theorem sletr

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

Ref Expression
Assertion sletr ( ( 𝐴 No 𝐵 No 𝐶 No ) → ( ( 𝐴 ≤s 𝐵𝐵 ≤s 𝐶 ) → 𝐴 ≤s 𝐶 ) )

Proof

Step Hyp Ref Expression
1 sltletr ( ( 𝐶 No 𝐴 No 𝐵 No ) → ( ( 𝐶 <s 𝐴𝐴 ≤s 𝐵 ) → 𝐶 <s 𝐵 ) )
2 1 3coml ( ( 𝐴 No 𝐵 No 𝐶 No ) → ( ( 𝐶 <s 𝐴𝐴 ≤s 𝐵 ) → 𝐶 <s 𝐵 ) )
3 2 expcomd ( ( 𝐴 No 𝐵 No 𝐶 No ) → ( 𝐴 ≤s 𝐵 → ( 𝐶 <s 𝐴𝐶 <s 𝐵 ) ) )
4 3 imp ( ( ( 𝐴 No 𝐵 No 𝐶 No ) ∧ 𝐴 ≤s 𝐵 ) → ( 𝐶 <s 𝐴𝐶 <s 𝐵 ) )
5 4 con3d ( ( ( 𝐴 No 𝐵 No 𝐶 No ) ∧ 𝐴 ≤s 𝐵 ) → ( ¬ 𝐶 <s 𝐵 → ¬ 𝐶 <s 𝐴 ) )
6 5 expimpd ( ( 𝐴 No 𝐵 No 𝐶 No ) → ( ( 𝐴 ≤s 𝐵 ∧ ¬ 𝐶 <s 𝐵 ) → ¬ 𝐶 <s 𝐴 ) )
7 slenlt ( ( 𝐵 No 𝐶 No ) → ( 𝐵 ≤s 𝐶 ↔ ¬ 𝐶 <s 𝐵 ) )
8 7 3adant1 ( ( 𝐴 No 𝐵 No 𝐶 No ) → ( 𝐵 ≤s 𝐶 ↔ ¬ 𝐶 <s 𝐵 ) )
9 8 anbi2d ( ( 𝐴 No 𝐵 No 𝐶 No ) → ( ( 𝐴 ≤s 𝐵𝐵 ≤s 𝐶 ) ↔ ( 𝐴 ≤s 𝐵 ∧ ¬ 𝐶 <s 𝐵 ) ) )
10 slenlt ( ( 𝐴 No 𝐶 No ) → ( 𝐴 ≤s 𝐶 ↔ ¬ 𝐶 <s 𝐴 ) )
11 10 3adant2 ( ( 𝐴 No 𝐵 No 𝐶 No ) → ( 𝐴 ≤s 𝐶 ↔ ¬ 𝐶 <s 𝐴 ) )
12 6 9 11 3imtr4d ( ( 𝐴 No 𝐵 No 𝐶 No ) → ( ( 𝐴 ≤s 𝐵𝐵 ≤s 𝐶 ) → 𝐴 ≤s 𝐶 ) )