Theorem oddneven

Description: An odd number is not an even number. (Contributed by AV, 16-Jun-2020)

	Ref	Expression
Assertion	oddneven	⊢ ( 𝑍 ∈ Odd → ¬ 𝑍 ∈ Even )

Proof

Step	Hyp	Ref	Expression
1		isodd	⊢ ( 𝑍 ∈ Odd ↔ ( 𝑍 ∈ ℤ ∧ ( ( 𝑍 + 1 ) / 2 ) ∈ ℤ ) )
2		zeo2	⊢ ( 𝑍 ∈ ℤ → ( ( 𝑍 / 2 ) ∈ ℤ ↔ ¬ ( ( 𝑍 + 1 ) / 2 ) ∈ ℤ ) )
3	2	biimpd	⊢ ( 𝑍 ∈ ℤ → ( ( 𝑍 / 2 ) ∈ ℤ → ¬ ( ( 𝑍 + 1 ) / 2 ) ∈ ℤ ) )
4	3	con2d	⊢ ( 𝑍 ∈ ℤ → ( ( ( 𝑍 + 1 ) / 2 ) ∈ ℤ → ¬ ( 𝑍 / 2 ) ∈ ℤ ) )
5	4	imp	⊢ ( ( 𝑍 ∈ ℤ ∧ ( ( 𝑍 + 1 ) / 2 ) ∈ ℤ ) → ¬ ( 𝑍 / 2 ) ∈ ℤ )
6	1 5	sylbi	⊢ ( 𝑍 ∈ Odd → ¬ ( 𝑍 / 2 ) ∈ ℤ )
7	6	olcd	⊢ ( 𝑍 ∈ Odd → ( ¬ 𝑍 ∈ ℤ ∨ ¬ ( 𝑍 / 2 ) ∈ ℤ ) )
8		ianor	⊢ ( ¬ ( 𝑍 ∈ ℤ ∧ ( 𝑍 / 2 ) ∈ ℤ ) ↔ ( ¬ 𝑍 ∈ ℤ ∨ ¬ ( 𝑍 / 2 ) ∈ ℤ ) )
9		iseven	⊢ ( 𝑍 ∈ Even ↔ ( 𝑍 ∈ ℤ ∧ ( 𝑍 / 2 ) ∈ ℤ ) )
10	8 9	xchnxbir	⊢ ( ¬ 𝑍 ∈ Even ↔ ( ¬ 𝑍 ∈ ℤ ∨ ¬ ( 𝑍 / 2 ) ∈ ℤ ) )
11	7 10	sylibr	⊢ ( 𝑍 ∈ Odd → ¬ 𝑍 ∈ Even )