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 } )