Theorem nnnn0i 12531

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 12530	. 2 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
3	1, 2	ax-mp 5	1 ⊢ 𝑁 ∈ ℕ0

Syntax hints:   ∈ wcel 2105  ℕcn 12263  ℕ₀cn0 12523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-v 3479  df-un 3967  df-ss 3979  df-n0 12524

This theorem is referenced by:  1nn0  12539  2nn0  12540  3nn0  12541  4nn0  12542  5nn0  12543  6nn0  12544  7nn0  12545  8nn0  12546  9nn0  12547  numlt  12755  declei  12766  numlti  12767  faclbnd4lem1  14328  divalglem6  16431  pockthi  16940  dec5dvds2  17098  modxp1i  17103  mod2xnegi  17104  43prm  17155  83prm  17156  317prm  17159  log2ublem2  27004  rpdp2cl2  32849  ballotlemfmpn  34475  ballotth  34518  circlevma  34635  12gcd5e1  41984  60gcd6e6  41985  60gcd7e1  41986  420lcm8e840  41992  lcmineqlem  42033  tgblthelfgott  47739  tgoldbach  47741