Theorem lenlteq

Description: 'less than or equal to' but not 'less than' implies 'equal' . (Contributed by Glauco Siliprandi, 3-Mar-2021)

	Ref	Expression
Hypotheses	lenlteq.1	|- ( ph -> A e. RR )
	lenlteq.2	|- ( ph -> B e. RR )
	lenlteq.3	|- ( ph -> A <_ B )
	lenlteq.4	|- ( ph -> -. A < B )
Assertion	lenlteq	|- ( ph -> A = B )

Proof

Step	Hyp	Ref	Expression
1		lenlteq.1	|- ( ph -> A e. RR )
2		lenlteq.2	|- ( ph -> B e. RR )
3		lenlteq.3	|- ( ph -> A <_ B )
4		lenlteq.4	|- ( ph -> -. A < B )
5	3 4	jca	|- ( ph -> ( A <_ B /\ -. A < B ) )
6		eqlelt	|- ( ( A e. RR /\ B e. RR ) -> ( A = B <-> ( A <_ B /\ -. A < B ) ) )
7	1 2 6	syl2anc	|- ( ph -> ( A = B <-> ( A <_ B /\ -. A < B ) ) )
8	5 7	mpbird	|- ( ph -> A = B )