Metamath Proof Explorer

Theorem sdomtr

Description: Strict dominance is transitive. Theorem 21(iii) of Suppes p. 97. (Contributed by NM, 9-Jun-1998)

Ref Expression
Assertion sdomtr 
|- ( ( A ~< B /\ B ~< C ) -> A ~< C )

Proof

Step Hyp Ref Expression
1 sdomdom 
 |-  ( A ~< B -> A ~<_ B )
2 domsdomtr 
 |-  ( ( A ~<_ B /\ B ~< C ) -> A ~< C )
3 1 2 sylan 
 |-  ( ( A ~< B /\ B ~< C ) -> A ~< C )