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| Mirrors > Home > MPE Home > Th. List > elnn0 | Structured version Visualization version GIF version | ||
| Description: Nonnegative integers expressed in terms of naturals and zero. (Contributed by Raph Levien, 10-Dec-2002.) |
| Ref | Expression |
|---|---|
| elnn0 | ⊢ (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-n0 12527 | . . 3 ⊢ ℕ0 = (ℕ ∪ {0}) | |
| 2 | 1 | eleq2i 2833 | . 2 ⊢ (𝐴 ∈ ℕ0 ↔ 𝐴 ∈ (ℕ ∪ {0})) |
| 3 | elun 4153 | . 2 ⊢ (𝐴 ∈ (ℕ ∪ {0}) ↔ (𝐴 ∈ ℕ ∨ 𝐴 ∈ {0})) | |
| 4 | c0ex 11255 | . . . 4 ⊢ 0 ∈ V | |
| 5 | 4 | elsn2 4665 | . . 3 ⊢ (𝐴 ∈ {0} ↔ 𝐴 = 0) |
| 6 | 5 | orbi2i 913 | . 2 ⊢ ((𝐴 ∈ ℕ ∨ 𝐴 ∈ {0}) ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0)) |
| 7 | 2, 3, 6 | 3bitri 297 | 1 ⊢ (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0)) |
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