Metamath Proof Explorer

Theorem sdomen1

Description: Equality-like theorem for equinumerosity and strict dominance. (Contributed by NM, 8-Nov-2003)

Ref Expression
Assertion sdomen1 
|- ( A ~~ B -> ( A ~< C <-> B ~< C ) )

Proof

Step Hyp Ref Expression
1 ensym 
 |-  ( A ~~ B -> B ~~ A )
2 ensdomtr 
 |-  ( ( B ~~ A /\ A ~< C ) -> B ~< C )
3 1 2 sylan 
 |-  ( ( A ~~ B /\ A ~< C ) -> B ~< C )
4 ensdomtr 
 |-  ( ( A ~~ B /\ B ~< C ) -> A ~< C )
5 3 4 impbida 
 |-  ( A ~~ B -> ( A ~< C <-> B ~< C ) )