Theorem xrltle

Description: 'Less than' implies 'less than or equal' for extended reals. (Contributed by NM, 19-Jan-2006)

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

Proof

Step	Hyp	Ref	Expression
1		orc	$$⊢A<B\to \left(A<B\vee A=B\right)$$
2		xrleloe	$$⊢\left(A\in {ℝ}^{*}\wedge B\in {ℝ}^{*}\right)\to \left(A\le B\leftrightarrow \left(A<B\vee A=B\right)\right)$$
3	1 2	syl5ibr	$$⊢\left(A\in {ℝ}^{*}\wedge B\in {ℝ}^{*}\right)\to \left(A<B\to A\le B\right)$$