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| Mirrors > Home > MPE Home > Th. List > nnz | Structured version Visualization version GIF version | ||
| Description: A positive integer is an integer. (Contributed by NM, 9-May-2004.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 29-Nov-2022.) |
| Ref | Expression |
|---|---|
| nnz | ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nnre 12273 | . 2 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
| 2 | 3mix2 1332 | . 2 ⊢ (𝑁 ∈ ℕ → (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)) | |
| 3 | elz 12615 | . 2 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ))) | |
| 4 | 1, 2, 3 | sylanbrc 583 | 1 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) |
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