Theorem sgnmnf

Description: The signum of -oo is -1. (Contributed by David A. Wheeler, 26-Jun-2016)

	Ref	Expression
Assertion	sgnmnf	$$⊢\mathrm{sgn}\left(\mathrm{-\infty }\right)=-1$$

Proof

Step	Hyp	Ref	Expression
1		mnfxr	$$⊢\mathrm{-\infty }\in {ℝ}^{*}$$
2		mnflt0	$$⊢\mathrm{-\infty }<0$$
3		sgnn	$$⊢\left(\mathrm{-\infty }\in {ℝ}^{*}\wedge \mathrm{-\infty }<0\right)\to \mathrm{sgn}\left(\mathrm{-\infty }\right)=-1$$
4	1 2 3	mp2an	$$⊢\mathrm{sgn}\left(\mathrm{-\infty }\right)=-1$$