| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > ltle | Structured version Visualization version GIF version | ||
| Description: 'Less than' implies 'less than or equal to'. (Contributed by NM, 25-Aug-1999.) |
| Ref | Expression |
|---|---|
| ltle | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → 𝐴 ≤ 𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | orc 868 | . 2 ⊢ (𝐴 < 𝐵 → (𝐴 < 𝐵 ∨ 𝐴 = 𝐵)) | |
| 2 | leloe 11347 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵))) | |
| 3 | 1, 2 | imbitrrid 246 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → 𝐴 ≤ 𝐵)) |
| Copyright terms: Public domain | W3C validator |