Theorem eqlei

Description: Equality implies 'less than or equal to'. (Contributed by NM, 23-May-1999) (Revised by Alexander van der Vekens, 20-Mar-2018)

	Ref	Expression
Hypothesis	lt.1	|- A e. RR
Assertion	eqlei	|- ( A = B -> A <_ B )

Proof

Step	Hyp	Ref	Expression
1		lt.1	|- A e. RR
2		eleq1a	|- ( A e. RR -> ( B = A -> B e. RR ) )
3	1 2	ax-mp	|- ( B = A -> B e. RR )
4	3	eqcoms	|- ( A = B -> B e. RR )
5		letri3	|- ( ( A e. RR /\ B e. RR ) -> ( A = B <-> ( A <_ B /\ B <_ A ) ) )
6	1 5	mpan	|- ( B e. RR -> ( A = B <-> ( A <_ B /\ B <_ A ) ) )
7		simpl	|- ( ( A <_ B /\ B <_ A ) -> A <_ B )
8	6 7	biimtrdi	|- ( B e. RR -> ( A = B -> A <_ B ) )
9	4 8	mpcom	|- ( A = B -> A <_ B )