| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > uzid | Structured version Visualization version GIF version | ||
| Description: Membership of the least member in an upper set of integers. (Contributed by NM, 2-Sep-2005.) |
| Ref | Expression |
|---|---|
| uzid | ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | id 22 | . 2 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℤ) | |
| 2 | zre 12617 | . . 3 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ) | |
| 3 | 2 | leidd 11829 | . 2 ⊢ (𝑀 ∈ ℤ → 𝑀 ≤ 𝑀) |
| 4 | eluz1 12882 | . 2 ⊢ (𝑀 ∈ ℤ → (𝑀 ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑀 ≤ 𝑀))) | |
| 5 | 1, 3, 4 | mpbir2and 713 | 1 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀)) |
| Copyright terms: Public domain | W3C validator |