Theorem nnnn0i 8189

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

Step	Hyp	Ref	Expression
1		nnnn0.1	. 2 ⊢ 𝑁 ∈ ℕ
2		nnnn0 8188	. 2 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
3	1, 2	ax-mp 7	1 ⊢ 𝑁 ∈ ℕ0

Syntax hints:   ∈ wcel 1393  ℕcn 7914  ℕ₀cn0 8181

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 8182

This theorem is referenced by:  1nn0  8197  2nn0  8198  3nn0  8199  4nn0  8200  5nn0  8201  6nn0  8202  7nn0  8203  8nn0  8204  9nn0  8205  10nn0  8206  numlt  8386  numlti  8391