Theorem evenz 47990

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 47988	. 2 ⊢ (𝑍 ∈ Even ↔ (𝑍 ∈ ℤ ∧ (𝑍 / 2) ∈ ℤ))
2	1	simplbi 496	1 ⊢ (𝑍 ∈ Even → 𝑍 ∈ ℤ)

Syntax hints:   → wi 4   ∈ wcel 2114  (class class class)co 7368   / cdiv 11806  2c2 12212  ℤcz 12500   Even ceven 47984

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6456  df-fv 6508  df-ov 7371  df-even 47986

This theorem is referenced by:  evenm1odd  47999  evenp1odd  48000  bits0eALTV  48040  opeoALTV  48044  omeoALTV  48046  epoo  48063  emoo  48064  epee  48065  emee  48066  evensumeven  48067  evenltle  48077  even3prm2  48079  mogoldbblem  48080  sbgoldbalt  48141  sgoldbeven3prm  48143  mogoldbb  48145  bgoldbachlt  48173  tgblthelfgott  48175