Theorem oddz 47113

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 47111	. 2 ⊢ (𝑍 ∈ Odd ↔ (𝑍 ∈ ℤ ∧ ((𝑍 + 1) / 2) ∈ ℤ))
2	1	simplbi 496	1 ⊢ (𝑍 ∈ Odd → 𝑍 ∈ ℤ)

Syntax hints:   → wi 4   ∈ wcel 2098  (class class class)co 7419  1c1 11146   + caddc 11148   / cdiv 11908  2c2 12305  ℤcz 12596   Odd codd 47107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-iota 6501  df-fv 6557  df-ov 7422  df-odd 47109

This theorem is referenced by:  oddm1div2z  47116  oddp1eveni  47123  oddm1eveni  47124  m1expoddALTV  47130  2dvdsoddp1  47138  2dvdsoddm1  47139  zofldiv2ALTV  47144  oddflALTV  47145  gcd2odd1  47150  oexpnegALTV  47159  oexpnegnz  47160  bits0oALTV  47163  opoeALTV  47165  opeoALTV  47166  omoeALTV  47167  omeoALTV  47168  epoo  47185  emoo  47186  stgoldbwt  47258  sbgoldbwt  47259  sbgoldbst  47260  sbgoldbm  47266  bgoldbtbndlem1  47287  bgoldbtbndlem2  47288  bgoldbtbndlem3  47289  bgoldbtbndlem4  47290  bgoldbtbnd  47291  tgoldbach  47299