Metamath Proof Explorer


Theorem suprlub

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

Ref Expression
Assertion suprlub ( ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦𝐴 𝑦𝑥 ) ∧ 𝐵 ∈ ℝ ) → ( 𝐵 < sup ( 𝐴 , ℝ , < ) ↔ ∃ 𝑧𝐴 𝐵 < 𝑧 ) )

Proof

Step Hyp Ref Expression
1 ltso < Or ℝ
2 1 a1i ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦𝐴 𝑦𝑥 ) → < Or ℝ )
3 sup3 ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦𝐴 𝑦𝑥 ) → ∃ 𝑥 ∈ ℝ ( ∀ 𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀ 𝑦 ∈ ℝ ( 𝑦 < 𝑥 → ∃ 𝑤𝐴 𝑦 < 𝑤 ) ) )
4 simp1 ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦𝐴 𝑦𝑥 ) → 𝐴 ⊆ ℝ )
5 2 3 4 suplub2 ( ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦𝐴 𝑦𝑥 ) ∧ 𝐵 ∈ ℝ ) → ( 𝐵 < sup ( 𝐴 , ℝ , < ) ↔ ∃ 𝑤𝐴 𝐵 < 𝑤 ) )
6 breq2 ( 𝑤 = 𝑧 → ( 𝐵 < 𝑤𝐵 < 𝑧 ) )
7 6 cbvrexvw ( ∃ 𝑤𝐴 𝐵 < 𝑤 ↔ ∃ 𝑧𝐴 𝐵 < 𝑧 )
8 5 7 bitrdi ( ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦𝐴 𝑦𝑥 ) ∧ 𝐵 ∈ ℝ ) → ( 𝐵 < sup ( 𝐴 , ℝ , < ) ↔ ∃ 𝑧𝐴 𝐵 < 𝑧 ) )