Metamath Proof Explorer


Theorem elmod2

Description: An integer modulo 2 is either 0 or 1. (Contributed by AV, 24-May-2020) (Proof shortened by OpenAI, 3-Jul-2020)

Ref Expression
Assertion elmod2 ( 𝑁 ∈ ℤ → ( 𝑁 mod 2 ) ∈ { 0 , 1 } )

Proof

Step Hyp Ref Expression
1 2nn 2 ∈ ℕ
2 zmodfzo ( ( 𝑁 ∈ ℤ ∧ 2 ∈ ℕ ) → ( 𝑁 mod 2 ) ∈ ( 0 ..^ 2 ) )
3 2 ancoms ( ( 2 ∈ ℕ ∧ 𝑁 ∈ ℤ ) → ( 𝑁 mod 2 ) ∈ ( 0 ..^ 2 ) )
4 1 3 mpan ( 𝑁 ∈ ℤ → ( 𝑁 mod 2 ) ∈ ( 0 ..^ 2 ) )
5 fzo0to2pr ( 0 ..^ 2 ) = { 0 , 1 }
6 4 5 eleqtrdi ( 𝑁 ∈ ℤ → ( 𝑁 mod 2 ) ∈ { 0 , 1 } )