Metamath Proof Explorer

Theorem max1

Description: A number is less than or equal to the maximum of it and another. See also max1ALT . (Contributed by NM, 3-Apr-2005)

Ref Expression
Assertion max1 
|- ( ( A e. RR /\ B e. RR ) -> A <_ if ( A <_ B , B , A ) )

Proof

Step Hyp Ref Expression
1 rexr 
 |-  ( A e. RR -> A e. RR* )
2 rexr 
 |-  ( B e. RR -> B e. RR* )
3 xrmax1 
 |-  ( ( A e. RR* /\ B e. RR* ) -> A <_ if ( A <_ B , B , A ) )
4 1 2 3 syl2an 
 |-  ( ( A e. RR /\ B e. RR ) -> A <_ if ( A <_ B , B , A ) )