Theorem oddz 47623

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 47621	. 2 ⊢ (𝑍 ∈ Odd ↔ (𝑍 ∈ ℤ ∧ ((𝑍 + 1) / 2) ∈ ℤ))
2	1	simplbi 497	1 ⊢ (𝑍 ∈ Odd → 𝑍 ∈ ℤ)

Syntax hints:   → wi 4   ∈ wcel 2107  (class class class)co 7432  1c1 11157   + caddc 11159   / cdiv 11921  2c2 12322  ℤcz 12615   Odd codd 47617

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-iota 6513  df-fv 6568  df-ov 7435  df-odd 47619

This theorem is referenced by:  oddm1div2z  47626  oddp1eveni  47633  oddm1eveni  47634  m1expoddALTV  47640  2dvdsoddp1  47648  2dvdsoddm1  47649  zofldiv2ALTV  47654  oddflALTV  47655  gcd2odd1  47660  oexpnegALTV  47669  oexpnegnz  47670  bits0oALTV  47673  opoeALTV  47675  opeoALTV  47676  omoeALTV  47677  omeoALTV  47678  epoo  47695  emoo  47696  stgoldbwt  47768  sbgoldbwt  47769  sbgoldbst  47770  sbgoldbm  47776  bgoldbtbndlem1  47797  bgoldbtbndlem2  47798  bgoldbtbndlem3  47799  bgoldbtbndlem4  47800  bgoldbtbnd  47801  tgoldbach  47809