Theorem leltned

Description: 'Less than or equal to' implies 'less than' is not 'equals'. (Contributed by Mario Carneiro, 27-May-2016)

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

Proof

Step	Hyp	Ref	Expression
1		ltd.1	|- ( ph -> A e. RR )
2		ltd.2	|- ( ph -> B e. RR )
3		leltned.3	|- ( ph -> A <_ B )
4		leltne	|- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( A < B <-> B =/= A ) )
5	1 2 3 4	syl3anc	|- ( ph -> ( A < B <-> B =/= A ) )