Theorem oddz 41544

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 41542	. 2 ⊢ (𝑍 ∈ Odd ↔ (𝑍 ∈ ℤ ∧ ((𝑍 + 1) / 2) ∈ ℤ))
2	1	simplbi 476	1 ⊢ (𝑍 ∈ Odd → 𝑍 ∈ ℤ)

Syntax hints:   → wi 4   ∈ wcel 1990  (class class class)co 6650  1c1 9937   + caddc 9939   / cdiv 10684  2c2 11070  ℤcz 11377   Odd codd 41538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-odd 41540

This theorem is referenced by:  oddm1div2z  41547  oddp1eveni  41554  oddm1eveni  41555  m1expoddALTV  41561  2dvdsoddp1  41568  2dvdsoddm1  41569  zofldiv2ALTV  41574  oddflALTV  41575  oexpnegALTV  41588  oexpnegnz  41589  bits0oALTV  41592  opoeALTV  41594  opeoALTV  41595  omoeALTV  41596  omeoALTV  41597  epoo  41612  emoo  41613  stgoldbwt  41664  sbgoldbwt  41665  sbgoldbst  41666  sbgoldbm  41672  bgoldbtbndlem1  41693  bgoldbtbndlem2  41694  bgoldbtbndlem3  41695  bgoldbtbndlem4  41696  bgoldbtbnd  41697  tgoldbach  41705  tgoldbachOLD  41712