Theorem nnnn0i 9521

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 9520	. 2 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
3	1, 2	ax-mp 5	1 ⊢ 𝑁 ∈ ℕ0

Syntax hints:   ∈ wcel 2205  ℕcn 9254  ℕ₀cn0 9513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-n0 9514

This theorem is referenced by:  1nn0  9529  2nn0  9530  3nn0  9531  4nn0  9532  5nn0  9533  6nn0  9534  7nn0  9535  8nn0  9536  9nn0  9537  numlt  9751  declei  9762  numlti  9763  pockthi  13081  dec5dvds2  13136  modxp1i  13141  ballotfilem1  13164  ballotfilemfmpn  13178  ballotfilemth  13225