This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 )