Theorem nnnn0i 12485

Description: A positive integer is a nonnegative integer. (Contributed by NM, 20-Jun-2005.)

Hypothesis

Ref	Expression
nnnn0i.1	⊢ 𝑁 ∈ ℕ

Assertion

Ref	Expression
nnnn0i	⊢ 𝑁 ∈ ℕ0

Proof of Theorem nnnn0i

Step	Hyp	Ref	Expression
1		nnnn0i.1	. 2 ⊢ 𝑁 ∈ ℕ
2		nnnn0 12484	. 2 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
3	1, 2	ax-mp 5	1 ⊢ 𝑁 ∈ ℕ0

Syntax hints:   ∈ wcel 2105  ℕcn 12217  ℕ₀cn0 12477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-un 3953  df-in 3955  df-ss 3965  df-n0 12478

This theorem is referenced by:  1nn0  12493  2nn0  12494  3nn0  12495  4nn0  12496  5nn0  12497  6nn0  12498  7nn0  12499  8nn0  12500  9nn0  12501  numlt  12707  declei  12718  numlti  12719  faclbnd4lem1  14258  divalglem6  16346  pockthi  16845  dec5dvds2  17003  modxp1i  17008  mod2xnegi  17009  43prm  17060  83prm  17061  317prm  17064  log2ublem2  26689  rpdp2cl2  32317  ballotlemfmpn  33792  ballotth  33835  circlevma  33953  12gcd5e1  41175  60gcd6e6  41176  60gcd7e1  41177  420lcm8e840  41183  lcmineqlem  41224  tgblthelfgott  46782  tgoldbach  46784