Theorem iseven 47874

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 7365	. . 3 ⊢ (𝑧 = 𝑍 → (𝑧 / 2) = (𝑍 / 2))
2	1	eleq1d 2821	. 2 ⊢ (𝑧 = 𝑍 → ((𝑧 / 2) ∈ ℤ ↔ (𝑍 / 2) ∈ ℤ))
3		df-even 47872	. 2 ⊢ Even = {𝑧 ∈ ℤ ∣ (𝑧 / 2) ∈ ℤ}
4	2, 3	elrab2 3649	1 ⊢ (𝑍 ∈ Even ↔ (𝑍 ∈ ℤ ∧ (𝑍 / 2) ∈ ℤ))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  (class class class)co 7358   / cdiv 11794  2c2 12200  ℤcz 12488   Even ceven 47870

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 2115  ax-9 2123  ax-ext 2708

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 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-iota 6448  df-fv 6500  df-ov 7361  df-even 47872

This theorem is referenced by:  evenz  47876  evendiv2z  47878  evenm1odd  47885  evenp1odd  47886  oddp1eveni  47887  oddm1eveni  47888  evennodd  47889  oddneven  47890  enege  47891  zeoALTV  47916  oddm1evenALTV  47921  oddp1evenALTV  47922  0evenALTV  47934  2evenALTV  47938  6even  47957  8even  47959