Theorem suprclii

Description: Closure of supremum of a nonempty bounded set of reals. (Contributed by NM, 12-Sep-1999)

	Ref	Expression
Hypothesis	sup3i.1	⊢ ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 )
Assertion	suprclii	⊢ sup ( 𝐴 , ℝ , < ) ∈ ℝ

Proof

Step	Hyp	Ref	Expression
1		sup3i.1	⊢ ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 )
2		suprcl	⊢ ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) → sup ( 𝐴 , ℝ , < ) ∈ ℝ )
3	1 2	ax-mp	⊢ sup ( 𝐴 , ℝ , < ) ∈ ℝ