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Metamath Proof Explorer

Theorem supxrcld

Description: The supremum of an arbitrary set of extended reals is an extended real. (Contributed by Glauco Siliprandi, 26-Jun-2021)

		Ref	Expression
	Hypothesis	supxrcld.1	$⊢ φ \to A \subseteq ℝ^{*}$
	Assertion	supxrcld	$⊢ φ \to sup (A ℝ^{} <) \in ℝ^{}$

Step	Hyp	Ref	Expression
1		supxrcld.1	$⊢ φ \to A \subseteq ℝ^{*}$
2		supxrcl	$⊢ A \subseteq ℝ^{} \to sup (A ℝ^{} <) \in ℝ^{*}$
3	1 2	syl	$⊢ φ \to sup (A ℝ^{} <) \in ℝ^{}$