Theorem supxrcli

Description: The supremum of an arbitrary set of extended reals is an extended real. (Contributed by Glauco Siliprandi, 2-Jan-2022)

	Ref	Expression
Hypothesis	supxrcli.1	⊢ 𝐴 ⊆ ℝ*
Assertion	supxrcli	⊢ sup ( 𝐴 , ℝ* , < ) ∈ ℝ*

Proof

Step	Hyp	Ref	Expression
1		supxrcli.1	⊢ 𝐴 ⊆ ℝ*
2		supxrcl	⊢ ( 𝐴 ⊆ ℝ* → sup ( 𝐴 , ℝ* , < ) ∈ ℝ* )
3	1 2	ax-mp	⊢ sup ( 𝐴 , ℝ* , < ) ∈ ℝ*