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| Mirrors > Home > MPE Home > Th. List > elfzoelz | Structured version Visualization version GIF version | ||
| Description: Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
| Ref | Expression |
|---|---|
| elfzoelz | ⊢ (𝐴 ∈ (𝐵..^𝐶) → 𝐴 ∈ ℤ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elfzoel1 13697 | . . . 4 ⊢ (𝐴 ∈ (𝐵..^𝐶) → 𝐵 ∈ ℤ) | |
| 2 | elfzoel2 13698 | . . . 4 ⊢ (𝐴 ∈ (𝐵..^𝐶) → 𝐶 ∈ ℤ) | |
| 3 | fzof 13696 | . . . . 5 ⊢ ..^:(ℤ × ℤ)⟶𝒫 ℤ | |
| 4 | 3 | fovcl 7561 | . . . 4 ⊢ ((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐵..^𝐶) ∈ 𝒫 ℤ) |
| 5 | 1, 2, 4 | syl2anc 584 | . . 3 ⊢ (𝐴 ∈ (𝐵..^𝐶) → (𝐵..^𝐶) ∈ 𝒫 ℤ) |
| 6 | 5 | elpwid 4609 | . 2 ⊢ (𝐴 ∈ (𝐵..^𝐶) → (𝐵..^𝐶) ⊆ ℤ) |
| 7 | id 22 | . 2 ⊢ (𝐴 ∈ (𝐵..^𝐶) → 𝐴 ∈ (𝐵..^𝐶)) | |
| 8 | 6, 7 | sseldd 3984 | 1 ⊢ (𝐴 ∈ (𝐵..^𝐶) → 𝐴 ∈ ℤ) |
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