Metamath Proof Explorer


Theorem bndndx

Description: A bounded real sequence A ( k ) is less than or equal to at least one of its indices. (Contributed by NM, 18-Jan-2008)

Ref Expression
Assertion bndndx ( ∃ 𝑥 ∈ ℝ ∀ 𝑘 ∈ ℕ ( 𝐴 ∈ ℝ ∧ 𝐴𝑥 ) → ∃ 𝑘 ∈ ℕ 𝐴𝑘 )

Proof

Step Hyp Ref Expression
1 arch ( 𝑥 ∈ ℝ → ∃ 𝑘 ∈ ℕ 𝑥 < 𝑘 )
2 nnre ( 𝑘 ∈ ℕ → 𝑘 ∈ ℝ )
3 lelttr ( ( 𝐴 ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ 𝑘 ∈ ℝ ) → ( ( 𝐴𝑥𝑥 < 𝑘 ) → 𝐴 < 𝑘 ) )
4 ltle ( ( 𝐴 ∈ ℝ ∧ 𝑘 ∈ ℝ ) → ( 𝐴 < 𝑘𝐴𝑘 ) )
5 4 3adant2 ( ( 𝐴 ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ 𝑘 ∈ ℝ ) → ( 𝐴 < 𝑘𝐴𝑘 ) )
6 3 5 syld ( ( 𝐴 ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ 𝑘 ∈ ℝ ) → ( ( 𝐴𝑥𝑥 < 𝑘 ) → 𝐴𝑘 ) )
7 6 exp5o ( 𝐴 ∈ ℝ → ( 𝑥 ∈ ℝ → ( 𝑘 ∈ ℝ → ( 𝐴𝑥 → ( 𝑥 < 𝑘𝐴𝑘 ) ) ) ) )
8 7 com3l ( 𝑥 ∈ ℝ → ( 𝑘 ∈ ℝ → ( 𝐴 ∈ ℝ → ( 𝐴𝑥 → ( 𝑥 < 𝑘𝐴𝑘 ) ) ) ) )
9 8 imp4b ( ( 𝑥 ∈ ℝ ∧ 𝑘 ∈ ℝ ) → ( ( 𝐴 ∈ ℝ ∧ 𝐴𝑥 ) → ( 𝑥 < 𝑘𝐴𝑘 ) ) )
10 9 com23 ( ( 𝑥 ∈ ℝ ∧ 𝑘 ∈ ℝ ) → ( 𝑥 < 𝑘 → ( ( 𝐴 ∈ ℝ ∧ 𝐴𝑥 ) → 𝐴𝑘 ) ) )
11 2 10 sylan2 ( ( 𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ ) → ( 𝑥 < 𝑘 → ( ( 𝐴 ∈ ℝ ∧ 𝐴𝑥 ) → 𝐴𝑘 ) ) )
12 11 reximdva ( 𝑥 ∈ ℝ → ( ∃ 𝑘 ∈ ℕ 𝑥 < 𝑘 → ∃ 𝑘 ∈ ℕ ( ( 𝐴 ∈ ℝ ∧ 𝐴𝑥 ) → 𝐴𝑘 ) ) )
13 1 12 mpd ( 𝑥 ∈ ℝ → ∃ 𝑘 ∈ ℕ ( ( 𝐴 ∈ ℝ ∧ 𝐴𝑥 ) → 𝐴𝑘 ) )
14 r19.35 ( ∃ 𝑘 ∈ ℕ ( ( 𝐴 ∈ ℝ ∧ 𝐴𝑥 ) → 𝐴𝑘 ) ↔ ( ∀ 𝑘 ∈ ℕ ( 𝐴 ∈ ℝ ∧ 𝐴𝑥 ) → ∃ 𝑘 ∈ ℕ 𝐴𝑘 ) )
15 13 14 sylib ( 𝑥 ∈ ℝ → ( ∀ 𝑘 ∈ ℕ ( 𝐴 ∈ ℝ ∧ 𝐴𝑥 ) → ∃ 𝑘 ∈ ℕ 𝐴𝑘 ) )
16 15 rexlimiv ( ∃ 𝑥 ∈ ℝ ∀ 𝑘 ∈ ℕ ( 𝐴 ∈ ℝ ∧ 𝐴𝑥 ) → ∃ 𝑘 ∈ ℕ 𝐴𝑘 )