Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 12091 | . . 3 ⊢ ℕ0 = (ℕ ∪ {0}) | |
2 | 1 | eleq2i 2829 | . 2 ⊢ (𝐴 ∈ ℕ0 ↔ 𝐴 ∈ (ℕ ∪ {0})) |
3 | elun 4063 | . 2 ⊢ (𝐴 ∈ (ℕ ∪ {0}) ↔ (𝐴 ∈ ℕ ∨ 𝐴 ∈ {0})) | |
4 | c0ex 10827 | . . . 4 ⊢ 0 ∈ V | |
5 | 4 | elsn2 4580 | . . 3 ⊢ (𝐴 ∈ {0} ↔ 𝐴 = 0) |
6 | 5 | orbi2i 913 | . 2 ⊢ ((𝐴 ∈ ℕ ∨ 𝐴 ∈ {0}) ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0)) |
7 | 2, 3, 6 | 3bitri 300 | 1 ⊢ (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0)) |
Copyright terms: Public domain | W3C validator |