Theorem nnssnn0 9196

Description: Positive naturals are a subset of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)

Assertion

Ref	Expression
nnssnn0	⊢ ℕ ⊆ ℕ0

Proof of Theorem nnssnn0

Step	Hyp	Ref	Expression
1		ssun1 3312	. 2 ⊢ ℕ ⊆ (ℕ ∪ {0})
2		df-n0 9194	. 2 ⊢ ℕ0 = (ℕ ∪ {0})
3	1, 2	sseqtrri 3204	1 ⊢ ℕ ⊆ ℕ0

Syntax hints:   ∪ cun 3141   ⊆ wss 3143  {csn 3606  0cc0 7828  ℕcn 8936  ℕ₀cn0 9193

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2170

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-v 2753  df-un 3147  df-in 3149  df-ss 3156  df-n0 9194

This theorem is referenced by:  nnnn0  9200  nnnn0d  9246  expcnv  11529  oddge22np1  11903