Theorem lttri3i

Description: Consequence of trichotomy. (Contributed by NM, 14-May-1999)

	Ref	Expression
Hypotheses	lt.1	$$⊢A\in ℝ$$
	lt.2	$$⊢B\in ℝ$$
Assertion	lttri3i	$$⊢A=B\leftrightarrow \left(\neg A<B\wedge \neg B<A\right)$$

Proof

Step	Hyp	Ref	Expression
1		lt.1	$$⊢A\in ℝ$$
2		lt.2	$$⊢B\in ℝ$$
3		lttri3	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to \left(A=B\leftrightarrow \left(\neg A<B\wedge \neg B<A\right)\right)$$
4	1 2 3	mp2an	$$⊢A=B\leftrightarrow \left(\neg A<B\wedge \neg B<A\right)$$