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Metamath Proof Explorer

Theorem suprleub

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
Assertion suprleub ( ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) ∧ 𝐵 ∈ ℝ ) → ( sup ( 𝐴 , ℝ , < ) ≤ 𝐵 ↔ ∀ 𝑧 ∈ 𝐴 𝑧 ≤ 𝐵 ) )

Proof

Step Hyp Ref Expression
1 suprnub ⊢ ( ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) ∧ 𝐵 ∈ ℝ ) → ( ¬ 𝐵 < sup ( 𝐴 , ℝ , < ) ↔ ∀ 𝑤 ∈ 𝐴 ¬ 𝐵 < 𝑤 ) )
2 suprcl ⊢ ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) → sup ( 𝐴 , ℝ , < ) ∈ ℝ )
3 lenlt ⊢ ( ( sup ( 𝐴 , ℝ , < ) ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( sup ( 𝐴 , ℝ , < ) ≤ 𝐵 ↔ ¬ 𝐵 < sup ( 𝐴 , ℝ , < ) ) )
4 2 3 sylan ⊢ ( ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) ∧ 𝐵 ∈ ℝ ) → ( sup ( 𝐴 , ℝ , < ) ≤ 𝐵 ↔ ¬ 𝐵 < sup ( 𝐴 , ℝ , < ) ) )
5 simpl1 ⊢ ( ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) ∧ 𝐵 ∈ ℝ ) → 𝐴 ⊆ ℝ )
6 5 sselda ⊢ ( ( ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) ∧ 𝐵 ∈ ℝ ) ∧ 𝑤 ∈ 𝐴 ) → 𝑤 ∈ ℝ )
7 simplr ⊢ ( ( ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) ∧ 𝐵 ∈ ℝ ) ∧ 𝑤 ∈ 𝐴 ) → 𝐵 ∈ ℝ )
8 6 7 lenltd ⊢ ( ( ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) ∧ 𝐵 ∈ ℝ ) ∧ 𝑤 ∈ 𝐴 ) → ( 𝑤 ≤ 𝐵 ↔ ¬ 𝐵 < 𝑤 ) )
9 8 ralbidva ⊢ ( ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) ∧ 𝐵 ∈ ℝ ) → ( ∀ 𝑤 ∈ 𝐴 𝑤 ≤ 𝐵 ↔ ∀ 𝑤 ∈ 𝐴 ¬ 𝐵 < 𝑤 ) )
10 1 4 9 3bitr4d ⊢ ( ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) ∧ 𝐵 ∈ ℝ ) → ( sup ( 𝐴 , ℝ , < ) ≤ 𝐵 ↔ ∀ 𝑤 ∈ 𝐴 𝑤 ≤ 𝐵 ) )
11 breq1 ⊢ ( 𝑤 = 𝑧 → ( 𝑤 ≤ 𝐵 ↔ 𝑧 ≤ 𝐵 ) )
12 11 cbvralvw ⊢ ( ∀ 𝑤 ∈ 𝐴 𝑤 ≤ 𝐵 ↔ ∀ 𝑧 ∈ 𝐴 𝑧 ≤ 𝐵 )
13 10 12 bitrdi ⊢ ( ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) ∧ 𝐵 ∈ ℝ ) → ( sup ( 𝐴 , ℝ , < ) ≤ 𝐵 ↔ ∀ 𝑧 ∈ 𝐴 𝑧 ≤ 𝐵 ) )