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	⊢ ¬ 2 ∥ 3

Proof

Step	Hyp	Ref	Expression
1		3halfnz	⊢ ¬ ( 3 / 2 ) ∈ ℤ
2		3z	⊢ 3 ∈ ℤ
3		evend2	⊢ ( 3 ∈ ℤ → ( 2 ∥ 3 ↔ ( 3 / 2 ) ∈ ℤ ) )
4	2 3	ax-mp	⊢ ( 2 ∥ 3 ↔ ( 3 / 2 ) ∈ ℤ )
5	1 4	mtbir	⊢ ¬ 2 ∥ 3