Metamath Proof Explorer

Theorem letric

Description: Trichotomy law. (Contributed by NM, 18-Aug-1999) (Proof shortened by Andrew Salmon, 19-Nov-2011)

Ref Expression
Assertion letric 
|- ( ( A e. RR /\ B e. RR ) -> ( A <_ B \/ B <_ A ) )

Proof

Step Hyp Ref Expression
1 ltnle 
 |-  ( ( B e. RR /\ A e. RR ) -> ( B < A <-> -. A <_ B ) )
2 ltle 
 |-  ( ( B e. RR /\ A e. RR ) -> ( B < A -> B <_ A ) )
3 1 2 sylbird 
 |-  ( ( B e. RR /\ A e. RR ) -> ( -. A <_ B -> B <_ A ) )
4 3 orrd 
 |-  ( ( B e. RR /\ A e. RR ) -> ( A <_ B \/ B <_ A ) )
5 4 ancoms 
 |-  ( ( A e. RR /\ B e. RR ) -> ( A <_ B \/ B <_ A ) )