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Theorem iseven 43786
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 43784 . 2 Even = {𝑧 ∈ ℤ ∣ (𝑧 / 2) ∈ ℤ}
42, 3elrab2 3683 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 43782
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 2156  ax-12 2172  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 3497  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-iota 6309  df-fv 6358  df-ov 7153  df-even 43784
This theorem is referenced by:  evenz  43788  evendiv2z  43790  evenm1odd  43797  evenp1odd  43798  oddp1eveni  43799  oddm1eveni  43800  evennodd  43801  oddneven  43802  enege  43803  zeoALTV  43828  oddm1evenALTV  43833  oddp1evenALTV  43834  0evenALTV  43846  2evenALTV  43850  6even  43869  8even  43871
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