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| Mirrors > Home > MPE Home > Th. List > elfzofz | Structured version Visualization version GIF version | ||
| Description: A half-open range is contained in the corresponding closed range. (Contributed by Stefan O'Rear, 23-Aug-2015.) |
| Ref | Expression |
|---|---|
| elfzofz | ⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝐾 ∈ (𝑀...𝑁)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elfzouz 13703 | . 2 ⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝐾 ∈ (ℤ≥‘𝑀)) | |
| 2 | elfzouz2 13714 | . 2 ⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝑁 ∈ (ℤ≥‘𝐾)) | |
| 3 | elfzuzb 13558 | . 2 ⊢ (𝐾 ∈ (𝑀...𝑁) ↔ (𝐾 ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ (ℤ≥‘𝐾))) | |
| 4 | 1, 2, 3 | sylanbrc 583 | 1 ⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝐾 ∈ (𝑀...𝑁)) |
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