Theorem iseven2

Description: The predicate "is an even number". An even number is an integer which is divisible by 2. (Contributed by AV, 18-Jun-2020)

	Ref	Expression
Assertion	iseven2	⊢ ( 𝑍 ∈ Even ↔ ( 𝑍 ∈ ℤ ∧ 2 ∥ 𝑍 ) )

Proof

Step	Hyp	Ref	Expression
1		breq2	⊢ ( 𝑧 = 𝑍 → ( 2 ∥ 𝑧 ↔ 2 ∥ 𝑍 ) )
2		dfeven2	⊢ Even = { 𝑧 ∈ ℤ ∣ 2 ∥ 𝑧 }
3	1 2	elrab2	⊢ ( 𝑍 ∈ Even ↔ ( 𝑍 ∈ ℤ ∧ 2 ∥ 𝑍 ) )