Theorem suprlubii

Description: The supremum of a nonempty bounded set of reals is the least upper bound. (Contributed by NM, 15-Oct-2004) (Revised by Mario Carneiro, 6-Sep-2014)

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

Proof

Step	Hyp	Ref	Expression
1		sup3i.1	⊢ ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 )
2		suprlub	⊢ ( ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) ∧ 𝐵 ∈ ℝ ) → ( 𝐵 < sup ( 𝐴 , ℝ , < ) ↔ ∃ 𝑧 ∈ 𝐴 𝐵 < 𝑧 ) )
3	1 2	mpan	⊢ ( 𝐵 ∈ ℝ → ( 𝐵 < sup ( 𝐴 , ℝ , < ) ↔ ∃ 𝑧 ∈ 𝐴 𝐵 < 𝑧 ) )