Theorem evenz 42568

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 42566	. 2 ⊢ (𝑍 ∈ Even ↔ (𝑍 ∈ ℤ ∧ (𝑍 / 2) ∈ ℤ))
2	1	simplbi 493	1 ⊢ (𝑍 ∈ Even → 𝑍 ∈ ℤ)

Syntax hints:   → wi 4   ∈ wcel 2107  (class class class)co 6922   / cdiv 11032  2c2 11430  ℤcz 11728   Even ceven 42562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-iota 6099  df-fv 6143  df-ov 6925  df-even 42564

This theorem is referenced by:  evenm1odd  42577  evenp1odd  42578  bits0eALTV  42616  opeoALTV  42620  omeoALTV  42622  epoo  42637  emoo  42638  epee  42639  emee  42640  evensumeven  42641  evenltle  42651  even3prm2  42653  mogoldbblem  42654  sbgoldbalt  42694  sgoldbeven3prm  42696  mogoldbb  42698  bgoldbachlt  42726  tgblthelfgott  42728