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| Mirrors > Home > MPE Home > Th. List > elfznn | Structured version Visualization version GIF version | ||
| Description: A member of a finite set of sequential integers starting at 1 is a positive integer. (Contributed by NM, 24-Aug-2005.) |
| Ref | Expression |
|---|---|
| elfznn | ⊢ (𝐾 ∈ (1...𝑁) → 𝐾 ∈ ℕ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elfzelz 13564 | . 2 ⊢ (𝐾 ∈ (1...𝑁) → 𝐾 ∈ ℤ) | |
| 2 | elfzle1 13567 | . 2 ⊢ (𝐾 ∈ (1...𝑁) → 1 ≤ 𝐾) | |
| 3 | elnnz1 12643 | . 2 ⊢ (𝐾 ∈ ℕ ↔ (𝐾 ∈ ℤ ∧ 1 ≤ 𝐾)) | |
| 4 | 1, 2, 3 | sylanbrc 583 | 1 ⊢ (𝐾 ∈ (1...𝑁) → 𝐾 ∈ ℕ) |
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