Metamath Proof Explorer


Theorem sotri

Description: A strict order relation is a transitive relation. (Contributed by NM, 10-Feb-1996) (Revised by Mario Carneiro, 10-May-2013)

Ref Expression
Hypotheses soi.1 𝑅 Or 𝑆
soi.2 𝑅 ⊆ ( 𝑆 × 𝑆 )
Assertion sotri ( ( 𝐴 𝑅 𝐵𝐵 𝑅 𝐶 ) → 𝐴 𝑅 𝐶 )

Proof

Step Hyp Ref Expression
1 soi.1 𝑅 Or 𝑆
2 soi.2 𝑅 ⊆ ( 𝑆 × 𝑆 )
3 2 brel ( 𝐴 𝑅 𝐵 → ( 𝐴𝑆𝐵𝑆 ) )
4 3 simpld ( 𝐴 𝑅 𝐵𝐴𝑆 )
5 2 brel ( 𝐵 𝑅 𝐶 → ( 𝐵𝑆𝐶𝑆 ) )
6 4 5 anim12i ( ( 𝐴 𝑅 𝐵𝐵 𝑅 𝐶 ) → ( 𝐴𝑆 ∧ ( 𝐵𝑆𝐶𝑆 ) ) )
7 sotr ( ( 𝑅 Or 𝑆 ∧ ( 𝐴𝑆𝐵𝑆𝐶𝑆 ) ) → ( ( 𝐴 𝑅 𝐵𝐵 𝑅 𝐶 ) → 𝐴 𝑅 𝐶 ) )
8 1 7 mpan ( ( 𝐴𝑆𝐵𝑆𝐶𝑆 ) → ( ( 𝐴 𝑅 𝐵𝐵 𝑅 𝐶 ) → 𝐴 𝑅 𝐶 ) )
9 8 3expb ( ( 𝐴𝑆 ∧ ( 𝐵𝑆𝐶𝑆 ) ) → ( ( 𝐴 𝑅 𝐵𝐵 𝑅 𝐶 ) → 𝐴 𝑅 𝐶 ) )
10 6 9 mpcom ( ( 𝐴 𝑅 𝐵𝐵 𝑅 𝐶 ) → 𝐴 𝑅 𝐶 )