Theorem iseven 47659

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.)

Assertion

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

Proof of Theorem iseven

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 7348	. . 3 ⊢ (𝑧 = 𝑍 → (𝑧 / 2) = (𝑍 / 2))
2	1	eleq1d 2816	. 2 ⊢ (𝑧 = 𝑍 → ((𝑧 / 2) ∈ ℤ ↔ (𝑍 / 2) ∈ ℤ))
3		df-even 47657	. 2 ⊢ Even = {𝑧 ∈ ℤ ∣ (𝑧 / 2) ∈ ℤ}
4	2, 3	elrab2 3645	1 ⊢ (𝑍 ∈ Even ↔ (𝑍 ∈ ℤ ∧ (𝑍 / 2) ∈ ℤ))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  (class class class)co 7341   / cdiv 11769  2c2 12175  ℤcz 12463   Even ceven 47655

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-iota 6432  df-fv 6484  df-ov 7344  df-even 47657

This theorem is referenced by:  evenz  47661  evendiv2z  47663  evenm1odd  47670  evenp1odd  47671  oddp1eveni  47672  oddm1eveni  47673  evennodd  47674  oddneven  47675  enege  47676  zeoALTV  47701  oddm1evenALTV  47706  oddp1evenALTV  47707  0evenALTV  47719  2evenALTV  47723  6even  47742  8even  47744