Theorem oddz 43803

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 43801	. 2 ⊢ (𝑍 ∈ Odd ↔ (𝑍 ∈ ℤ ∧ ((𝑍 + 1) / 2) ∈ ℤ))
2	1	simplbi 500	1 ⊢ (𝑍 ∈ Odd → 𝑍 ∈ ℤ)

Syntax hints:   → wi 4   ∈ wcel 2114  (class class class)co 7158  1c1 10540   + caddc 10542   / cdiv 11299  2c2 11695  ℤcz 11984   Odd codd 43797

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-iota 6316  df-fv 6365  df-ov 7161  df-odd 43799

This theorem is referenced by:  oddm1div2z  43806  oddp1eveni  43813  oddm1eveni  43814  m1expoddALTV  43820  2dvdsoddp1  43828  2dvdsoddm1  43829  zofldiv2ALTV  43834  oddflALTV  43835  gcd2odd1  43840  oexpnegALTV  43849  oexpnegnz  43850  bits0oALTV  43853  opoeALTV  43855  opeoALTV  43856  omoeALTV  43857  omeoALTV  43858  epoo  43875  emoo  43876  stgoldbwt  43948  sbgoldbwt  43949  sbgoldbst  43950  sbgoldbm  43956  bgoldbtbndlem1  43977  bgoldbtbndlem2  43978  bgoldbtbndlem3  43979  bgoldbtbndlem4  43980  bgoldbtbnd  43981  tgoldbach  43989