Theorem min1d

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

	Ref	Expression
Hypotheses	min1d.1	|- ( ph -> A e. RR )
	min1d.2	|- ( ph -> B e. RR )
Assertion	min1d	|- ( ph -> if ( A <_ B , A , B ) <_ A )

Proof

Step	Hyp	Ref	Expression
1		min1d.1	|- ( ph -> A e. RR )
2		min1d.2	|- ( ph -> B e. RR )
3		min1	|- ( ( A e. RR /\ B e. RR ) -> if ( A <_ B , A , B ) <_ A )
4	1 2 3	syl2anc	|- ( ph -> if ( A <_ B , A , B ) <_ A )