Metamath Proof Explorer

Theorem lttri4

Description: Trichotomy law for 'less than'. (Contributed by NM, 20-Sep-2007) (Proof shortened by Andrew Salmon, 19-Nov-2011)

Ref Expression
Assertion lttri4 
|- ( ( A e. RR /\ B e. RR ) -> ( A < B \/ A = B \/ B < A ) )

Proof

Step Hyp Ref Expression
1 ltso 
 |-  < Or RR
2 solin 
 |-  ( ( < Or RR /\ ( A e. RR /\ B e. RR ) ) -> ( A < B \/ A = B \/ B < A ) )
3 1 2 mpan 
 |-  ( ( A e. RR /\ B e. RR ) -> ( A < B \/ A = B \/ B < A ) )