Metamath Proof Explorer


Theorem max2d

Description: A number is less than or equal to the maximum of it and another. (Contributed by Glauco Siliprandi, 2-Jan-2022)

Ref Expression
Hypotheses max2d.1
|- ( ph -> A e. RR )
max2d.2
|- ( ph -> B e. RR )
Assertion max2d
|- ( ph -> B <_ if ( A <_ B , B , A ) )

Proof

Step Hyp Ref Expression
1 max2d.1
 |-  ( ph -> A e. RR )
2 max2d.2
 |-  ( ph -> B e. RR )
3 max2
 |-  ( ( A e. RR /\ B e. RR ) -> B <_ if ( A <_ B , B , A ) )
4 1 2 3 syl2anc
 |-  ( ph -> B <_ if ( A <_ B , B , A ) )