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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 ∧ 𝐴 ≤ 𝑥 ) → ∃ 𝑘 ∈ ℕ 𝐴 ≤ 𝑘 )

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 ) → ( ( 𝐴 ≤ 𝑥 ∧ 𝑥 < 𝑘 ) → 𝐴 < 𝑘 ) )
4 ltle ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑘 ∈ ℝ ) → ( 𝐴 < 𝑘 → 𝐴 ≤ 𝑘 ) )
5 4 3adant2 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ 𝑘 ∈ ℝ ) → ( 𝐴 < 𝑘 → 𝐴 ≤ 𝑘 ) )
6 3 5 syld ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ 𝑘 ∈ ℝ ) → ( ( 𝐴 ≤ 𝑥 ∧ 𝑥 < 𝑘 ) → 𝐴 ≤ 𝑘 ) )
7 6 exp5o ⊢ ( 𝐴 ∈ ℝ → ( 𝑥 ∈ ℝ → ( 𝑘 ∈ ℝ → ( 𝐴 ≤ 𝑥 → ( 𝑥 < 𝑘 → 𝐴 ≤ 𝑘 ) ) ) ) )
8 7 com3l ⊢ ( 𝑥 ∈ ℝ → ( 𝑘 ∈ ℝ → ( 𝐴 ∈ ℝ → ( 𝐴 ≤ 𝑥 → ( 𝑥 < 𝑘 → 𝐴 ≤ 𝑘 ) ) ) ) )
9 8 imp4b ⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑘 ∈ ℝ ) → ( ( 𝐴 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ) → ( 𝑥 < 𝑘 → 𝐴 ≤ 𝑘 ) ) )
10 9 com23 ⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑘 ∈ ℝ ) → ( 𝑥 < 𝑘 → ( ( 𝐴 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ) → 𝐴 ≤ 𝑘 ) ) )
11 2 10 sylan2 ⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ ) → ( 𝑥 < 𝑘 → ( ( 𝐴 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ) → 𝐴 ≤ 𝑘 ) ) )
12 11 reximdva ⊢ ( 𝑥 ∈ ℝ → ( ∃ 𝑘 ∈ ℕ 𝑥 < 𝑘 → ∃ 𝑘 ∈ ℕ ( ( 𝐴 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ) → 𝐴 ≤ 𝑘 ) ) )
13 1 12 mpd ⊢ ( 𝑥 ∈ ℝ → ∃ 𝑘 ∈ ℕ ( ( 𝐴 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ) → 𝐴 ≤ 𝑘 ) )
14 r19.35 ⊢ ( ∃ 𝑘 ∈ ℕ ( ( 𝐴 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ) → 𝐴 ≤ 𝑘 ) ↔ ( ∀ 𝑘 ∈ ℕ ( 𝐴 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ) → ∃ 𝑘 ∈ ℕ 𝐴 ≤ 𝑘 ) )
15 13 14 sylib ⊢ ( 𝑥 ∈ ℝ → ( ∀ 𝑘 ∈ ℕ ( 𝐴 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ) → ∃ 𝑘 ∈ ℕ 𝐴 ≤ 𝑘 ) )
16 15 rexlimiv ⊢ ( ∃ 𝑥 ∈ ℝ ∀ 𝑘 ∈ ℕ ( 𝐴 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ) → ∃ 𝑘 ∈ ℕ 𝐴 ≤ 𝑘 )