Theorem ltnltne

Description: Variant of trichotomy law for 'less than'. (Contributed by Alexander van der Vekens, 8-Jun-2018)

	Ref	Expression
Assertion	ltnltne	|- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> ( -. B < A /\ -. B = A ) ) )

Proof

Step	Hyp	Ref	Expression
1		ltnle	|- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> -. B <_ A ) )
2		leloe	|- ( ( B e. RR /\ A e. RR ) -> ( B <_ A <-> ( B < A \/ B = A ) ) )
3	2	ancoms	|- ( ( A e. RR /\ B e. RR ) -> ( B <_ A <-> ( B < A \/ B = A ) ) )
4	3	notbid	|- ( ( A e. RR /\ B e. RR ) -> ( -. B <_ A <-> -. ( B < A \/ B = A ) ) )
5		ioran	|- ( -. ( B < A \/ B = A ) <-> ( -. B < A /\ -. B = A ) )
6	5	a1i	|- ( ( A e. RR /\ B e. RR ) -> ( -. ( B < A \/ B = A ) <-> ( -. B < A /\ -. B = A ) ) )
7	1 4 6	3bitrd	|- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> ( -. B < A /\ -. B = A ) ) )