Theorem oddz 47505

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 47503	. 2 ⊢ (𝑍 ∈ Odd ↔ (𝑍 ∈ ℤ ∧ ((𝑍 + 1) / 2) ∈ ℤ))
2	1	simplbi 497	1 ⊢ (𝑍 ∈ Odd → 𝑍 ∈ ℤ)

Syntax hints:   → wi 4   ∈ wcel 2108  (class class class)co 7448  1c1 11185   + caddc 11187   / cdiv 11947  2c2 12348  ℤcz 12639   Odd codd 47499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6525  df-fv 6581  df-ov 7451  df-odd 47501

This theorem is referenced by:  oddm1div2z  47508  oddp1eveni  47515  oddm1eveni  47516  m1expoddALTV  47522  2dvdsoddp1  47530  2dvdsoddm1  47531  zofldiv2ALTV  47536  oddflALTV  47537  gcd2odd1  47542  oexpnegALTV  47551  oexpnegnz  47552  bits0oALTV  47555  opoeALTV  47557  opeoALTV  47558  omoeALTV  47559  omeoALTV  47560  epoo  47577  emoo  47578  stgoldbwt  47650  sbgoldbwt  47651  sbgoldbst  47652  sbgoldbm  47658  bgoldbtbndlem1  47679  bgoldbtbndlem2  47680  bgoldbtbndlem3  47681  bgoldbtbndlem4  47682  bgoldbtbnd  47683  tgoldbach  47691