Theorem xrinfm

Description: The extended real numbers are unbounded below. (Contributed by Thierry Arnoux, 18-Feb-2018) (Revised by AV, 28-Sep-2020)

	Ref	Expression
Assertion	xrinfm	$$⊢sup\left({ℝ}^{*},{ℝ}^{*},<\right)=\mathrm{-\infty }$$

Proof

Step	Hyp	Ref	Expression
1		ssid	$$⊢{ℝ}^{*}\subseteq {ℝ}^{*}$$
2		mnfxr	$$⊢\mathrm{-\infty }\in {ℝ}^{*}$$
3		infxrmnf	$$⊢\left({ℝ}^{*}\subseteq {ℝ}^{*}\wedge \mathrm{-\infty }\in {ℝ}^{*}\right)\to sup\left({ℝ}^{*},{ℝ}^{*},<\right)=\mathrm{-\infty }$$
4	1 2 3	mp2an	$$⊢sup\left({ℝ}^{*},{ℝ}^{*},<\right)=\mathrm{-\infty }$$