Theorem oddz 45141

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 45139	. 2 ⊢ (𝑍 ∈ Odd ↔ (𝑍 ∈ ℤ ∧ ((𝑍 + 1) / 2) ∈ ℤ))
2	1	simplbi 499	1 ⊢ (𝑍 ∈ Odd → 𝑍 ∈ ℤ)

Syntax hints:   → wi 4   ∈ wcel 2104  (class class class)co 7307  1c1 10918   + caddc 10920   / cdiv 11678  2c2 12074  ℤcz 12365   Odd codd 45135

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3287  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-br 5082  df-iota 6410  df-fv 6466  df-ov 7310  df-odd 45137

This theorem is referenced by:  oddm1div2z  45144  oddp1eveni  45151  oddm1eveni  45152  m1expoddALTV  45158  2dvdsoddp1  45166  2dvdsoddm1  45167  zofldiv2ALTV  45172  oddflALTV  45173  gcd2odd1  45178  oexpnegALTV  45187  oexpnegnz  45188  bits0oALTV  45191  opoeALTV  45193  opeoALTV  45194  omoeALTV  45195  omeoALTV  45196  epoo  45213  emoo  45214  stgoldbwt  45286  sbgoldbwt  45287  sbgoldbst  45288  sbgoldbm  45294  bgoldbtbndlem1  45315  bgoldbtbndlem2  45316  bgoldbtbndlem3  45317  bgoldbtbndlem4  45318  bgoldbtbnd  45319  tgoldbach  45327