Metamath Proof Explorer

Theorem oddp1evenALTV

Description: An integer is odd iff its successor is even. (Contributed by Mario Carneiro, 5-Sep-2016) (Revised by AV, 19-Jun-2020)

Ref Expression
Assertion oddp1evenALTV ⊢N∈ℤ→N∈Odd↔N+1∈Even

Proof

Step Hyp Ref Expression
1 isodd ⊢N∈Odd↔N∈ℤ∧N+12∈ℤ
2 1 baib ⊢N∈ℤ→N∈Odd↔N+12∈ℤ
3 peano2z ⊢N∈ℤ→N+1∈ℤ
4 3 biantrurd ⊢N∈ℤ→N+12∈ℤ↔N+1∈ℤ∧N+12∈ℤ
5 2 4 bitrd ⊢N∈ℤ→N∈Odd↔N+1∈ℤ∧N+12∈ℤ
6 iseven ⊢N+1∈Even↔N+1∈ℤ∧N+12∈ℤ
7 5 6 bitr4di ⊢N∈ℤ→N∈Odd↔N+1∈Even