Theorem n2dvds3

Description: 2 does not divide 3. That means 3 is odd. (Contributed by AV, 28-Feb-2021) (Proof shortened by Steven Nguyen, 3-May-2023)

	Ref	Expression
Assertion	n2dvds3	$$⊢\neg 2\parallel 3$$

Proof

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