Metamath Proof Explorer

Theorem ltlei

Description: 'Less than' implies 'less than or equal to'. (Contributed by NM, 14-May-1999)

Ref Expression
Hypotheses lt.1 
|- A e. RR
lt.2 
|- B e. RR
Assertion ltlei 
|- ( A < B -> A <_ B )

Proof

Step Hyp Ref Expression
1 lt.1 
 |-  A e. RR
2 lt.2 
 |-  B e. RR
3 ltle 
 |-  ( ( A e. RR /\ B e. RR ) -> ( A < B -> A <_ B ) )
4 1 2 3 mp2an 
 |-  ( A < B -> A <_ B )