Theorem letric

Description: Trichotomy law. (Contributed by NM, 18-Aug-1999) (Proof shortened by Andrew Salmon, 19-Nov-2011)

	Ref	Expression
Assertion	letric	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to \left(A\le B\vee B\le A\right)$$

Proof

Step	Hyp	Ref	Expression
1		ltnle	$$⊢\left(B\in ℝ\wedge A\in ℝ\right)\to \left(B<A\leftrightarrow \neg A\le B\right)$$
2		ltle	$$⊢\left(B\in ℝ\wedge A\in ℝ\right)\to \left(B<A\to B\le A\right)$$
3	1 2	sylbird	$$⊢\left(B\in ℝ\wedge A\in ℝ\right)\to \left(\neg A\le B\to B\le A\right)$$
4	3	orrd	$$⊢\left(B\in ℝ\wedge A\in ℝ\right)\to \left(A\le B\vee B\le A\right)$$
5	4	ancoms	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to \left(A\le B\vee B\le A\right)$$