Theorem xrleneltd

Description: 'Less than or equal to' and 'not equals' implies 'less than', for extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020)

	Ref	Expression
Hypotheses	xrleneltd.a	|- ( ph -> A e. RR* )
	xrleneltd.b	|- ( ph -> B e. RR* )
	xrleneltd.alb	|- ( ph -> A <_ B )
	xrleneltd.anb	|- ( ph -> A =/= B )
Assertion	xrleneltd	|- ( ph -> A < B )

Proof

Step	Hyp	Ref	Expression
1		xrleneltd.a	|- ( ph -> A e. RR* )
2		xrleneltd.b	|- ( ph -> B e. RR* )
3		xrleneltd.alb	|- ( ph -> A <_ B )
4		xrleneltd.anb	|- ( ph -> A =/= B )
5	4	necomd	|- ( ph -> B =/= A )
6		xrleltne	|- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> ( A < B <-> B =/= A ) )
7	1 2 3 6	syl3anc	|- ( ph -> ( A < B <-> B =/= A ) )
8	5 7	mpbird	|- ( ph -> A < B )