Description: A positive integer is even or odd. (Contributed by AV, 30-May-2020)
Ref | Expression | ||
---|---|---|---|
Assertion | nneop | ⊢ ( 𝑁 ∈ ℕ → ( ( 𝑁 / 2 ) ∈ ℕ ∨ ( ( 𝑁 + 1 ) / 2 ) ∈ ℕ ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nneo | ⊢ ( 𝑁 ∈ ℕ → ( ( 𝑁 / 2 ) ∈ ℕ ↔ ¬ ( ( 𝑁 + 1 ) / 2 ) ∈ ℕ ) ) | |
2 | 1 | biimprd | ⊢ ( 𝑁 ∈ ℕ → ( ¬ ( ( 𝑁 + 1 ) / 2 ) ∈ ℕ → ( 𝑁 / 2 ) ∈ ℕ ) ) |
3 | 2 | orrd | ⊢ ( 𝑁 ∈ ℕ → ( ( ( 𝑁 + 1 ) / 2 ) ∈ ℕ ∨ ( 𝑁 / 2 ) ∈ ℕ ) ) |
4 | 3 | orcomd | ⊢ ( 𝑁 ∈ ℕ → ( ( 𝑁 / 2 ) ∈ ℕ ∨ ( ( 𝑁 + 1 ) / 2 ) ∈ ℕ ) ) |