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Theorem nnnn0 8246
Description: A positive integer is a nonnegative integer. (Contributed by NM, 9-May-2004.)
Assertion
Ref Expression
nnnn0 (𝐴 ∈ ℕ → 𝐴 ∈ ℕ0)

Proof of Theorem nnnn0
StepHypRef Expression
1 nnssnn0 8242 . 2 ℕ ⊆ ℕ0
21sseli 2969 1 (𝐴 ∈ ℕ → 𝐴 ∈ ℕ0)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1409  cn 7990  0cn0 8239
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-n0 8240
This theorem is referenced by:  nnnn0i  8247  elnnnn0b  8283  elnnnn0c  8284  elnn0z  8315  elz2  8370  nn0ind-raph  8414  zindd  8415  fzo1fzo0n0  9141  ubmelfzo  9158  elfzom1elp1fzo  9160  fzo0sn0fzo1  9179  modqmulnn  9292  expnegap0  9428  expcllem  9431  expcl2lemap  9432  expap0  9450  expeq0  9451  mulexpzap  9460  expnlbnd  9541  facdiv  9606  faclbnd  9609  faclbnd3  9611  faclbnd6  9612  resqrexlemlo  9840  absexpzap  9907  nn0enne  10214  nnehalf  10216  nno  10218  nn0o  10219  divalg2  10238  ndvdssub  10242  pw2dvds  10254  oddpwdc  10262
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