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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 ) )