Theorem iseven 40079

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		df-even 40077	. . 3 ⊢ Even = {𝑧 ∈ ℤ ∣ (𝑧 / 2) ∈ ℤ}
2	1	eleq2i 2680	. 2 ⊢ (𝑍 ∈ Even ↔ 𝑍 ∈ {𝑧 ∈ ℤ ∣ (𝑧 / 2) ∈ ℤ})
3		oveq1 6556	. . . 4 ⊢ (𝑧 = 𝑍 → (𝑧 / 2) = (𝑍 / 2))
4	3	eleq1d 2672	. . 3 ⊢ (𝑧 = 𝑍 → ((𝑧 / 2) ∈ ℤ ↔ (𝑍 / 2) ∈ ℤ))
5	4	elrab 3331	. 2 ⊢ (𝑍 ∈ {𝑧 ∈ ℤ ∣ (𝑧 / 2) ∈ ℤ} ↔ (𝑍 ∈ ℤ ∧ (𝑍 / 2) ∈ ℤ))
6	2, 5	bitri 263	1 ⊢ (𝑍 ∈ Even ↔ (𝑍 ∈ ℤ ∧ (𝑍 / 2) ∈ ℤ))

Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900  (class class class)co 6549   / cdiv 10563  2c2 10947  ℤcz 11254   Even ceven 40075

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-even 40077

This theorem is referenced by:  evenz  40081  evendiv2z  40083  evenm1odd  40090  evenp1odd  40091  oddp1eveni  40092  oddm1eveni  40093  evennodd  40094  oddneven  40095  enege  40096  zeoALTV  40119  oddm1evenALTV  40124  oddp1evenALTV  40125  0evenALTV  40137  2evenALTV  40141  6even  40158  8even  40160