Metamath Proof Explorer


Theorem iseven

Description: The predicate "is an even number". An even number is an integer which is divisible by 2, i.e. the result of dividing the even integer by 2 is still an integer. (Contributed by AV, 14-Jun-2020)

Ref Expression
Assertion iseven ( 𝑍 ∈ Even ↔ ( 𝑍 ∈ ℤ ∧ ( 𝑍 / 2 ) ∈ ℤ ) )

Proof

Step Hyp Ref Expression
1 oveq1 ( 𝑧 = 𝑍 → ( 𝑧 / 2 ) = ( 𝑍 / 2 ) )
2 1 eleq1d ( 𝑧 = 𝑍 → ( ( 𝑧 / 2 ) ∈ ℤ ↔ ( 𝑍 / 2 ) ∈ ℤ ) )
3 df-even Even = { 𝑧 ∈ ℤ ∣ ( 𝑧 / 2 ) ∈ ℤ }
4 2 3 elrab2 ( 𝑍 ∈ Even ↔ ( 𝑍 ∈ ℤ ∧ ( 𝑍 / 2 ) ∈ ℤ ) )