Metamath Proof Explorer

Theorem min1

Description: The minimum of two numbers is less than or equal to the first. (Contributed by NM, 3-Aug-2007)

Ref Expression
Assertion min1 
|- ( ( A e. RR /\ B e. RR ) -> if ( A <_ B , A , B ) <_ A )

Proof

Step Hyp Ref Expression
1 rexr 
 |-  ( A e. RR -> A e. RR* )
2 rexr 
 |-  ( B e. RR -> B e. RR* )
3 xrmin1 
 |-  ( ( A e. RR* /\ B e. RR* ) -> if ( A <_ B , A , B ) <_ A )
4 1 2 3 syl2an 
 |-  ( ( A e. RR /\ B e. RR ) -> if ( A <_ B , A , B ) <_ A )