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Theorem iseven 43787
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.
StepHypRef Expression
1 oveq1 7157 . . 3 (𝑧 = 𝑍 → (𝑧 / 2) = (𝑍 / 2))
21eleq1d 2897 . 2 (𝑧 = 𝑍 → ((𝑧 / 2) ∈ ℤ ↔ (𝑍 / 2) ∈ ℤ))
3 df-even 43785 . 2 Even = {𝑧 ∈ ℤ ∣ (𝑧 / 2) ∈ ℤ}
42, 3elrab2 3682 1 (𝑍 ∈ Even ↔ (𝑍 ∈ ℤ ∧ (𝑍 / 2) ∈ ℤ))
Colors of variables: wff setvar class
Syntax hints:  wb 208  wa 398   = wceq 1533  wcel 2110  (class class class)co 7150   / cdiv 11291  2c2 11686  cz 11975   Even ceven 43783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-iota 6308  df-fv 6357  df-ov 7153  df-even 43785
This theorem is referenced by:  evenz  43789  evendiv2z  43791  evenm1odd  43798  evenp1odd  43799  oddp1eveni  43800  oddm1eveni  43801  evennodd  43802  oddneven  43803  enege  43804  zeoALTV  43829  oddm1evenALTV  43834  oddp1evenALTV  43835  0evenALTV  43847  2evenALTV  43851  6even  43870  8even  43872
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