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 e. ZZ -> ( N mod 2 ) e. { 0 , 1 } )

Proof

Step	Hyp	Ref	Expression
1		2nn	|- 2 e. NN
2		zmodfzo	|- ( ( N e. ZZ /\ 2 e. NN ) -> ( N mod 2 ) e. ( 0 ..^ 2 ) )
3	2	ancoms	|- ( ( 2 e. NN /\ N e. ZZ ) -> ( N mod 2 ) e. ( 0 ..^ 2 ) )
4	1 3	mpan	|- ( N e. ZZ -> ( N mod 2 ) e. ( 0 ..^ 2 ) )
5		fzo0to2pr	|- ( 0 ..^ 2 ) = { 0 , 1 }
6	4 5	eleqtrdi	|- ( N e. ZZ -> ( N mod 2 ) e. { 0 , 1 } )