Metamath Proof Explorer

Theorem xrltle

Description: 'Less than' implies 'less than or equal' for extended reals. (Contributed by NM, 19-Jan-2006)

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

Proof

Step Hyp Ref Expression
1 orc 
 |-  ( A < B -> ( A < B \/ A = B ) )
2 xrleloe 
 |-  ( ( A e. RR* /\ B e. RR* ) -> ( A <_ B <-> ( A < B \/ A = B ) ) )
3 1 2 syl5ibr 
 |-  ( ( A e. RR* /\ B e. RR* ) -> ( A < B -> A <_ B ) )