Theorem lttri2i

Description: Consequence of trichotomy. (Contributed by NM, 19-Jan-1997)

	Ref	Expression
Hypotheses	lt.1	$$⊢A\in ℝ$$
	lt.2	$$⊢B\in ℝ$$
Assertion	lttri2i	$$⊢A\ne B\leftrightarrow \left(A<B\vee B<A\right)$$

Proof

Step	Hyp	Ref	Expression
1		lt.1	$$⊢A\in ℝ$$
2		lt.2	$$⊢B\in ℝ$$
3		lttri2	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to \left(A\ne B\leftrightarrow \left(A<B\vee B<A\right)\right)$$
4	1 2 3	mp2an	$$⊢A\ne B\leftrightarrow \left(A<B\vee B<A\right)$$