Theorem evenz 47644

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 47642	. 2 ⊢ (𝑍 ∈ Even ↔ (𝑍 ∈ ℤ ∧ (𝑍 / 2) ∈ ℤ))
2	1	simplbi 497	1 ⊢ (𝑍 ∈ Even → 𝑍 ∈ ℤ)

Syntax hints:   → wi 4   ∈ wcel 2108  (class class class)co 7405   / cdiv 11894  2c2 12295  ℤcz 12588   Even ceven 47638

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-iota 6484  df-fv 6539  df-ov 7408  df-even 47640

This theorem is referenced by:  evenm1odd  47653  evenp1odd  47654  bits0eALTV  47694  opeoALTV  47698  omeoALTV  47700  epoo  47717  emoo  47718  epee  47719  emee  47720  evensumeven  47721  evenltle  47731  even3prm2  47733  mogoldbblem  47734  sbgoldbalt  47795  sgoldbeven3prm  47797  mogoldbb  47799  bgoldbachlt  47827  tgblthelfgott  47829