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Theorem nnnn0 8186
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 8182 . 2 ℕ ⊆ ℕ0
21sseli 2941 1 (𝐴 ∈ ℕ → 𝐴 ∈ ℕ0)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1393  cn 7912  0cn0 8179
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-n0 8180
This theorem is referenced by:  nnnn0i  8187  elnnnn0b  8224  elnnnn0c  8225  elnn0z  8256  elz2  8310  nn0ind-raph  8353  zindd  8354  fzo1fzo0n0  9037  ubmelfzo  9054  elfzom1elp1fzo  9056  fzo0sn0fzo1  9075  expnegap0  9237  expcllem  9240  expcl2lemap  9241  expap0  9259  expeq0  9260  mulexpzap  9269  expnlbnd  9347  resqrexlemlo  9585  absexpzap  9650
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