Theorem nnssnn0 9117

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 3285	. 2 ⊢ ℕ ⊆ (ℕ ∪ {0})
2		df-n0 9115	. 2 ⊢ ℕ0 = (ℕ ∪ {0})
3	1, 2	sseqtrri 3177	1 ⊢ ℕ ⊆ ℕ0

Syntax hints:   ∪ cun 3114   ⊆ wss 3116  {csn 3576  0cc0 7753  ℕcn 8857  ℕ₀cn0 9114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-n0 9115

This theorem is referenced by:  nnnn0  9121  nnnn0d  9167  expcnv  11445  oddge22np1  11818