Theorem lttrid

Description: Ordering on reals satisfies strict trichotomy. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	ltd.1	|- ( ph -> A e. RR )
	ltd.2	|- ( ph -> B e. RR )
Assertion	lttrid	|- ( 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		axlttri	|- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> -. ( A = B \/ B < A ) ) )
4	1 2 3	syl2anc	|- ( ph -> ( A < B <-> -. ( A = B \/ B < A ) ) )