Theorem evennodd

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

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

Proof

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