Metamath Proof Explorer

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	$⊢ (A \subseteq ℝ^{} \land −∞ \in A) \to sup (A ℝ^{} <) = −∞$

Proof

Step	Hyp	Ref	Expression
1		infxrlb	$⊢ (A \subseteq ℝ^{} \land −∞ \in A) \to sup (A ℝ^{} <) \leq −∞$
2		infxrcl	$⊢ A \subseteq ℝ^{} \to sup (A ℝ^{} <) \in ℝ^{*}$
3	2	adantr	$⊢ (A \subseteq ℝ^{} \land −∞ \in A) \to sup (A ℝ^{} <) \in ℝ^{*}$
4		xlemnf	$⊢ sup (A ℝ^{} <) \in ℝ^{} \to (sup (A ℝ^{} <) \leq −∞ \leftrightarrow sup (A ℝ^{} <) = −∞)$
5	3 4	syl	$⊢ (A \subseteq ℝ^{} \land −∞ \in A) \to (sup (A ℝ^{} <) \leq −∞ \leftrightarrow sup (A ℝ^{*} <) = −∞)$
6	1 5	mpbid	$⊢ (A \subseteq ℝ^{} \land −∞ \in A) \to sup (A ℝ^{} <) = −∞$