Theorem nnssnn0 9360

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 3367	. 2 ⊢ ℕ ⊆ (ℕ ∪ {0})
2		df-n0 9358	. 2 ⊢ ℕ0 = (ℕ ∪ {0})
3	1, 2	sseqtrri 3259	1 ⊢ ℕ ⊆ ℕ0

Syntax hints:   ∪ cun 3195   ⊆ wss 3197  {csn 3666  0cc0 7987  ℕcn 9098  ℕ₀cn0 9357

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-n0 9358

This theorem is referenced by:  nnnn0  9364  nnnn0d  9410  expcnv  12001  oddge22np1  12378  bitsfzolem  12451