Theorem infxrmnf

Description: The infinimum of a set of extended reals containing minus infinity is minus infinity. (Contributed by Thierry Arnoux, 18-Feb-2018) (Revised by AV, 28-Sep-2020)

	Ref	Expression
Assertion	infxrmnf	$$⊢\left(A\subseteq {ℝ}^{*}\wedge \mathrm{-\infty }\in A\right)\to sup\left(A,{ℝ}^{*},<\right)=\mathrm{-\infty }$$

Proof

Step	Hyp	Ref	Expression
1		infxrlb	$$⊢\left(A\subseteq {ℝ}^{*}\wedge \mathrm{-\infty }\in A\right)\to sup\left(A,{ℝ}^{*},<\right)\le \mathrm{-\infty }$$
2		infxrcl	$$⊢A\subseteq {ℝ}^{*}\to sup\left(A,{ℝ}^{*},<\right)\in {ℝ}^{*}$$
3	2	adantr	$$⊢\left(A\subseteq {ℝ}^{*}\wedge \mathrm{-\infty }\in A\right)\to sup\left(A,{ℝ}^{*},<\right)\in {ℝ}^{*}$$
4		xlemnf	$$⊢sup\left(A,{ℝ}^{*},<\right)\in {ℝ}^{*}\to \left(sup\left(A,{ℝ}^{*},<\right)\le \mathrm{-\infty }\leftrightarrow sup\left(A,{ℝ}^{*},<\right)=\mathrm{-\infty }\right)$$
5	3 4	syl	$$⊢\left(A\subseteq {ℝ}^{*}\wedge \mathrm{-\infty }\in A\right)\to \left(sup\left(A,{ℝ}^{*},<\right)\le \mathrm{-\infty }\leftrightarrow sup\left(A,{ℝ}^{*},<\right)=\mathrm{-\infty }\right)$$
6	1 5	mpbid	$$⊢\left(A\subseteq {ℝ}^{*}\wedge \mathrm{-\infty }\in A\right)\to sup\left(A,{ℝ}^{*},<\right)=\mathrm{-\infty }$$