Theorem lttri3

Description: Trichotomy law for 'less than'. (Contributed by NM, 5-May-1999)

	Ref	Expression
Assertion	lttri3	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to \left(A=B\leftrightarrow \left(\neg A<B\wedge \neg B<A\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		ltso	$$⊢<\mathrm{Or}ℝ$$
2		sotrieq2	$$⊢\left(<\mathrm{Or}ℝ\wedge \left(A\in ℝ\wedge B\in ℝ\right)\right)\to \left(A=B\leftrightarrow \left(\neg A<B\wedge \neg B<A\right)\right)$$
3	1 2	mpan	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to \left(A=B\leftrightarrow \left(\neg A<B\wedge \neg B<A\right)\right)$$