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	$$⊢\phi \to A\subseteq {ℝ}^{*}$$
Assertion	supxrcld	$$⊢\phi \to sup\left(A,{ℝ}^{*},<\right)\in {ℝ}^{*}$$

Proof

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