Theorem leneltd

Description: 'Less than or equal to' and 'not equals' implies 'less than'. (Contributed by Glauco Siliprandi, 11-Dec-2019)

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

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		leneltd.4	|- ( ph -> B =/= A )
5	1 2 3	leltned	|- ( ph -> ( A < B <-> B =/= A ) )
6	4 5	mpbird	|- ( ph -> A < B )