Theorem lttri3d

Description: Consequence of trichotomy. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	ltd.1	|- ( ph -> A e. RR )
	ltd.2	|- ( ph -> B e. RR )
Assertion	lttri3d	|- ( ph -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )

Proof

Step	Hyp	Ref	Expression
1		ltd.1	|- ( ph -> A e. RR )
2		ltd.2	|- ( ph -> B e. RR )
3		lttri3	|- ( ( A e. RR /\ B e. RR ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
4	1 2 3	syl2anc	|- ( ph -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )