Metamath Proof Explorer


Theorem min2d

Description: The minimum of two numbers is less than or equal to the second. (Contributed by Glauco Siliprandi, 5-Feb-2022)

Ref Expression
Hypotheses min2d.1
|- ( ph -> A e. RR )
min2d.2
|- ( ph -> B e. RR )
Assertion min2d
|- ( ph -> if ( A <_ B , A , B ) <_ B )

Proof

Step Hyp Ref Expression
1 min2d.1
 |-  ( ph -> A e. RR )
2 min2d.2
 |-  ( ph -> B e. RR )
3 min2
 |-  ( ( A e. RR /\ B e. RR ) -> if ( A <_ B , A , B ) <_ B )
4 1 2 3 syl2anc
 |-  ( ph -> if ( A <_ B , A , B ) <_ B )