Theorem xrlelttric

Description: Trichotomy law for extended reals. (Contributed by Thierry Arnoux, 12-Sep-2017)

	Ref	Expression
Assertion	xrlelttric	$$⊢\left(A\in {ℝ}^{*}\wedge B\in {ℝ}^{*}\right)\to \left(A\le B\vee B<A\right)$$

Proof

Step	Hyp	Ref	Expression
1		pm2.1	$$⊢\left(\neg B<A\vee B<A\right)$$
2		xrlenlt	$$⊢\left(A\in {ℝ}^{*}\wedge B\in {ℝ}^{*}\right)\to \left(A\le B\leftrightarrow \neg B<A\right)$$
3	2	orbi1d	$$⊢\left(A\in {ℝ}^{*}\wedge B\in {ℝ}^{*}\right)\to \left(\left(A\le B\vee B<A\right)\leftrightarrow \left(\neg B<A\vee B<A\right)\right)$$
4	1 3	mpbiri	$$⊢\left(A\in {ℝ}^{*}\wedge B\in {ℝ}^{*}\right)\to \left(A\le B\vee B<A\right)$$