Theorem xrlenltd

Description: "Less than or equal to" expressed in terms of "less than", for extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020)

	Ref	Expression
Hypotheses	xrlenltd.a	$$⊢\phi \to A\in {ℝ}^{*}$$
	xrlenltd.b	$$⊢\phi \to B\in {ℝ}^{*}$$
Assertion	xrlenltd	$$⊢\phi \to \left(A\le B\leftrightarrow \neg B<A\right)$$

Proof

Step	Hyp	Ref	Expression
1		xrlenltd.a	$$⊢\phi \to A\in {ℝ}^{*}$$
2		xrlenltd.b	$$⊢\phi \to B\in {ℝ}^{*}$$
3		xrlenlt	$$⊢\left(A\in {ℝ}^{*}\wedge B\in {ℝ}^{*}\right)\to \left(A\le B\leftrightarrow \neg B<A\right)$$
4	1 2 3	syl2anc	$$⊢\phi \to \left(A\le B\leftrightarrow \neg B<A\right)$$