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Metamath Proof Explorer

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 )