Theorem suprleubii

Description: The supremum of a nonempty bounded set of reals is less than or equal to an upper bound. (Contributed by NM, 18-Mar-2005) (Revised by Mario Carneiro, 6-Sep-2014)

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

Proof

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