Theorem nnnn0i 9504

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 9503	. 2 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
3	1, 2	ax-mp 5	1 ⊢ 𝑁 ∈ ℕ0

Syntax hints:   ∈ wcel 2203  ℕcn 9237  ℕ₀cn0 9496

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 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-n0 9497

This theorem is referenced by:  1nn0  9512  2nn0  9513  3nn0  9514  4nn0  9515  5nn0  9516  6nn0  9517  7nn0  9518  8nn0  9519  9nn0  9520  numlt  9733  declei  9744  numlti  9745  pockthi  13056  dec5dvds2  13111  modxp1i  13116  ballotfilem1  13139