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Metamath Proof Explorer

Theorem n2dvds1

Description: 2 does not divide 1. That means 1 is odd. (Contributed by David A. Wheeler, 8-Dec-2018) (Proof shortened by Steven Nguyen, 3-May-2023)

		Ref	Expression
	Assertion	n2dvds1	$⊢ \neg 2 ∥ 1$

Proof

Step	Hyp	Ref	Expression
1		halfnz	$⊢ \neg \frac{1}{2} \in ℤ$
2		1z	$⊢ 1 \in ℤ$
3		evend2	$⊢ 1 \in ℤ \to (2 ∥ 1 \leftrightarrow \frac{1}{2} \in ℤ)$
4	2 3	ax-mp	$⊢ 2 ∥ 1 \leftrightarrow \frac{1}{2} \in ℤ$
5	1 4	mtbir	$⊢ \neg 2 ∥ 1$