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Metamath Proof Explorer

Theorem zeo4

Description: An integer is even or odd but not both. With this representation of even and odd integers, this variant of zeo2 follows immediately from the principle of double negation, see notnotb . (Contributed by AV, 17-Jun-2021)

		Ref	Expression
	Assertion	zeo4	\|- ( N e. ZZ -> ( 2 \|\| N <-> -. -. 2 \|\| N ) )

Proof

Step	Hyp	Ref	Expression
1		notnotb	\|- ( 2 \|\| N <-> -. -. 2 \|\| N )
2	1	a1i	\|- ( N e. ZZ -> ( 2 \|\| N <-> -. -. 2 \|\| N ) )