Metamath Proof Explorer

Theorem ensdomtr

Description: Transitivity of equinumerosity and strict dominance. (Contributed by NM, 26-Oct-2003) (Revised by Mario Carneiro, 26-Apr-2015)

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

Proof

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