Theorem evenz 47876

Description: An even number is an integer. (Contributed by AV, 14-Jun-2020.)

Assertion

Ref	Expression
evenz	⊢ (𝑍 ∈ Even → 𝑍 ∈ ℤ)

Proof of Theorem evenz

Step	Hyp	Ref	Expression
1		iseven 47874	. 2 ⊢ (𝑍 ∈ Even ↔ (𝑍 ∈ ℤ ∧ (𝑍 / 2) ∈ ℤ))
2	1	simplbi 497	1 ⊢ (𝑍 ∈ Even → 𝑍 ∈ ℤ)

Syntax hints:   → wi 4   ∈ 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:  evenm1odd  47885  evenp1odd  47886  bits0eALTV  47926  opeoALTV  47930  omeoALTV  47932  epoo  47949  emoo  47950  epee  47951  emee  47952  evensumeven  47953  evenltle  47963  even3prm2  47965  mogoldbblem  47966  sbgoldbalt  48027  sgoldbeven3prm  48029  mogoldbb  48031  bgoldbachlt  48059  tgblthelfgott  48061