Theorem oddz 46299

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 46297	. 2 ⊢ (𝑍 ∈ Odd ↔ (𝑍 ∈ ℤ ∧ ((𝑍 + 1) / 2) ∈ ℤ))
2	1	simplbi 499	1 ⊢ (𝑍 ∈ Odd → 𝑍 ∈ ℤ)

Syntax hints:   → wi 4   ∈ wcel 2107  (class class class)co 7409  1c1 11111   + caddc 11113   / cdiv 11871  2c2 12267  ℤcz 12558   Odd codd 46293

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-ov 7412  df-odd 46295

This theorem is referenced by:  oddm1div2z  46302  oddp1eveni  46309  oddm1eveni  46310  m1expoddALTV  46316  2dvdsoddp1  46324  2dvdsoddm1  46325  zofldiv2ALTV  46330  oddflALTV  46331  gcd2odd1  46336  oexpnegALTV  46345  oexpnegnz  46346  bits0oALTV  46349  opoeALTV  46351  opeoALTV  46352  omoeALTV  46353  omeoALTV  46354  epoo  46371  emoo  46372  stgoldbwt  46444  sbgoldbwt  46445  sbgoldbst  46446  sbgoldbm  46452  bgoldbtbndlem1  46473  bgoldbtbndlem2  46474  bgoldbtbndlem3  46475  bgoldbtbndlem4  46476  bgoldbtbnd  46477  tgoldbach  46485