Theorem evenz 49001

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 48999	. 2 ⊢ (𝑍 ∈ Even ↔ (𝑍 ∈ ℤ ∧ (𝑍 / 2) ∈ ℤ))
2	1	simplbi 502	1 ⊢ (𝑍 ∈ Even → 𝑍 ∈ ℤ)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147  (class class class)co 7451   / cdiv 12013  2c2 12437  ℤcz 12733   Even ceven 48995

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-iota 6520  df-fv 6574  df-ov 7454  df-even 48997

This theorem is used by:  evenm1odd  49010  evenp1odd  49011  bits0eALTV  49051  opeoALTV  49055  omeoALTV  49057  epoo  49074  emoo  49075  epee  49076  emee  49077  evensumeven  49078  evenltle  49088  even3prm2  49090  mogoldbblem  49091  sbgoldbalt  49152  sgoldbeven3prm  49154  mogoldbb  49156  bgoldbachlt  49184  tgblthelfgott  49186