Theorem infxrcld

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

	Ref	Expression
Hypothesis	infxrcld.1	⊢ ( 𝜑 → 𝐴 ⊆ ℝ* )
Assertion	infxrcld	⊢ ( 𝜑 → inf ( 𝐴 , ℝ* , < ) ∈ ℝ* )

Proof

Step	Hyp	Ref	Expression
1		infxrcld.1	⊢ ( 𝜑 → 𝐴 ⊆ ℝ* )
2		infxrcl	⊢ ( 𝐴 ⊆ ℝ* → inf ( 𝐴 , ℝ* , < ) ∈ ℝ* )
3	1 2	syl	⊢ ( 𝜑 → inf ( 𝐴 , ℝ* , < ) ∈ ℝ* )