Theorem suprubii

Description: A member of a nonempty bounded set of reals is less than or equal to the set's upper bound. (Contributed by NM, 12-Sep-1999)

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

Proof

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