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Theorem nnnn0i 8992
 Description: A positive integer is a nonnegative integer. (Contributed by NM, 20-Jun-2005.)
Hypothesis
Ref Expression
nnnn0.1 𝑁 ∈ ℕ
Assertion
Ref Expression
nnnn0i 𝑁 ∈ ℕ0

Proof of Theorem nnnn0i
StepHypRef Expression
1 nnnn0.1 . 2 𝑁 ∈ ℕ
2 nnnn0 8991 . 2 (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
31, 2ax-mp 5 1 𝑁 ∈ ℕ0
 Colors of variables: wff set class Syntax hints:   ∈ wcel 1480  ℕcn 8727  ℕ0cn0 8984 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-n0 8985 This theorem is referenced by:  1nn0  9000  2nn0  9001  3nn0  9002  4nn0  9003  5nn0  9004  6nn0  9005  7nn0  9006  8nn0  9007  9nn0  9008  numlt  9213  declei  9224  numlti  9225
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