Theorem iseven 44146

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 7142	. . 3 ⊢ (𝑧 = 𝑍 → (𝑧 / 2) = (𝑍 / 2))
2	1	eleq1d 2874	. 2 ⊢ (𝑧 = 𝑍 → ((𝑧 / 2) ∈ ℤ ↔ (𝑍 / 2) ∈ ℤ))
3		df-even 44144	. 2 ⊢ Even = {𝑧 ∈ ℤ ∣ (𝑧 / 2) ∈ ℤ}
4	2, 3	elrab2 3631	1 ⊢ (𝑍 ∈ Even ↔ (𝑍 ∈ ℤ ∧ (𝑍 / 2) ∈ ℤ))

Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  (class class class)co 7135   / cdiv 11286  2c2 11680  ℤcz 11969   Even ceven 44142

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-rab 3115  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-iota 6283  df-fv 6332  df-ov 7138  df-even 44144

This theorem is referenced by:  evenz  44148  evendiv2z  44150  evenm1odd  44157  evenp1odd  44158  oddp1eveni  44159  oddm1eveni  44160  evennodd  44161  oddneven  44162  enege  44163  zeoALTV  44188  oddm1evenALTV  44193  oddp1evenALTV  44194  0evenALTV  44206  2evenALTV  44210  6even  44229  8even  44231