Theorem evenz 48453

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 48451	. 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 2146  (class class class)co 7419   / cdiv 11886  2c2 12310  ℤcz 12606   Even ceven 48447

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

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 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-iota 6496  df-fv 6548  df-ov 7422  df-even 48449

This theorem is used by:  evenm1odd  48462  evenp1odd  48463  bits0eALTV  48503  opeoALTV  48507  omeoALTV  48509  epoo  48526  emoo  48527  epee  48528  emee  48529  evensumeven  48530  evenltle  48540  even3prm2  48542  mogoldbblem  48543  sbgoldbalt  48604  sgoldbeven3prm  48606  mogoldbb  48608  bgoldbachlt  48636  tgblthelfgott  48638