Theorem infrecl

Description: Closure of infimum of a nonempty bounded set of reals. (Contributed by NM, 8-Oct-2005) (Revised by AV, 4-Sep-2020)

	Ref	Expression
Assertion	infrecl	⊢ ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ) → inf ( 𝐴 , ℝ , < ) ∈ ℝ )

Proof

Step	Hyp	Ref	Expression
1		ltso	⊢ < Or ℝ
2	1	a1i	⊢ ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ) → < Or ℝ )
3		infm3	⊢ ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ) → ∃ 𝑥 ∈ ℝ ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀ 𝑦 ∈ ℝ ( 𝑥 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) )
4	2 3	infcl	⊢ ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ) → inf ( 𝐴 , ℝ , < ) ∈ ℝ )