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 → 𝐴 ⊆ ℝ* )
Assertion	supxrcld	⊢ ( 𝜑 → sup ( 𝐴 , ℝ* , < ) ∈ ℝ* )

Proof

Step	Hyp	Ref	Expression
1		supxrcld.1	⊢ ( 𝜑 → 𝐴 ⊆ ℝ* )
2		supxrcl	⊢ ( 𝐴 ⊆ ℝ* → sup ( 𝐴 , ℝ* , < ) ∈ ℝ* )
3	1 2	syl	⊢ ( 𝜑 → sup ( 𝐴 , ℝ* , < ) ∈ ℝ* )