Metamath Proof Explorer


Theorem nneop

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 ) )

Proof

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 ) )