Theorem lttri4

Description: Trichotomy law for 'less than'. (Contributed by NM, 20-Sep-2007) (Proof shortened by Andrew Salmon, 19-Nov-2011)

	Ref	Expression
Assertion	lttri4	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to \left(A<B\vee A=B\vee B<A\right)$$

Proof

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