Theorem xrltnle

Description: "Less than" expressed in terms of "less than or equal to", for extended reals. (Contributed by NM, 6-Feb-2007)

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

Proof

Step	Hyp	Ref	Expression
1		xrlenlt	$$⊢\left(B\in {ℝ}^{*}\wedge A\in {ℝ}^{*}\right)\to \left(B\le A\leftrightarrow \neg A<B\right)$$
2	1	con2bid	$$⊢\left(B\in {ℝ}^{*}\wedge A\in {ℝ}^{*}\right)\to \left(A<B\leftrightarrow \neg B\le A\right)$$
3	2	ancoms	$$⊢\left(A\in {ℝ}^{*}\wedge B\in {ℝ}^{*}\right)\to \left(A<B\leftrightarrow \neg B\le A\right)$$