Theorem nnnn0i 12389

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 12388	. 2 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
3	1, 2	ax-mp 5	1 ⊢ 𝑁 ∈ ℕ0

Syntax hints:   ∈ wcel 2111  ℕcn 12125  ℕ₀cn0 12381

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 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-un 3902  df-ss 3914  df-n0 12382

This theorem is referenced by:  1nn0  12397  2nn0  12398  3nn0  12399  4nn0  12400  5nn0  12401  6nn0  12402  7nn0  12403  8nn0  12404  9nn0  12405  numlt  12613  declei  12624  numlti  12625  faclbnd4lem1  14200  divalglem6  16309  pockthi  16819  dec5dvds2  16977  modxp1i  16982  mod2xnegi  16983  43prm  17033  83prm  17034  317prm  17037  log2ublem2  26884  rpdp2cl2  32863  ballotlemfmpn  34508  ballotth  34551  circlevma  34655  12gcd5e1  42106  60gcd6e6  42107  60gcd7e1  42108  420lcm8e840  42114  lcmineqlem  42155  tgblthelfgott  47925  tgoldbach  47927