Theorem iseven 47553

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 7438	. . 3 ⊢ (𝑧 = 𝑍 → (𝑧 / 2) = (𝑍 / 2))
2	1	eleq1d 2824	. 2 ⊢ (𝑧 = 𝑍 → ((𝑧 / 2) ∈ ℤ ↔ (𝑍 / 2) ∈ ℤ))
3		df-even 47551	. 2 ⊢ Even = {𝑧 ∈ ℤ ∣ (𝑧 / 2) ∈ ℤ}
4	2, 3	elrab2 3698	1 ⊢ (𝑍 ∈ Even ↔ (𝑍 ∈ ℤ ∧ (𝑍 / 2) ∈ ℤ))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106  (class class class)co 7431   / cdiv 11918  2c2 12319  ℤcz 12611   Even ceven 47549

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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-even 47551

This theorem is referenced by:  evenz  47555  evendiv2z  47557  evenm1odd  47564  evenp1odd  47565  oddp1eveni  47566  oddm1eveni  47567  evennodd  47568  oddneven  47569  enege  47570  zeoALTV  47595  oddm1evenALTV  47600  oddp1evenALTV  47601  0evenALTV  47613  2evenALTV  47617  6even  47636  8even  47638