Theorem oddz 47793

Description: An odd number is an integer. (Contributed by AV, 14-Jun-2020.)

Assertion

Ref	Expression
oddz	⊢ (𝑍 ∈ Odd → 𝑍 ∈ ℤ)

Proof of Theorem oddz

Step	Hyp	Ref	Expression
1		isodd 47791	. 2 ⊢ (𝑍 ∈ Odd ↔ (𝑍 ∈ ℤ ∧ ((𝑍 + 1) / 2) ∈ ℤ))
2	1	simplbi 497	1 ⊢ (𝑍 ∈ Odd → 𝑍 ∈ ℤ)

Syntax hints:   → wi 4   ∈ wcel 2113  (class class class)co 7355  1c1 11018   + caddc 11020   / cdiv 11785  2c2 12191  ℤcz 12479   Odd codd 47787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-iota 6445  df-fv 6497  df-ov 7358  df-odd 47789

This theorem is referenced by:  oddm1div2z  47796  oddp1eveni  47803  oddm1eveni  47804  m1expoddALTV  47810  2dvdsoddp1  47818  2dvdsoddm1  47819  zofldiv2ALTV  47824  oddflALTV  47825  gcd2odd1  47830  oexpnegALTV  47839  oexpnegnz  47840  bits0oALTV  47843  opoeALTV  47845  opeoALTV  47846  omoeALTV  47847  omeoALTV  47848  epoo  47865  emoo  47866  stgoldbwt  47938  sbgoldbwt  47939  sbgoldbst  47940  sbgoldbm  47946  bgoldbtbndlem1  47967  bgoldbtbndlem2  47968  bgoldbtbndlem3  47969  bgoldbtbndlem4  47970  bgoldbtbnd  47971  tgoldbach  47979