Theorem nnssnn0 9405

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 3370	. 2 ⊢ ℕ ⊆ (ℕ ∪ {0})
2		df-n0 9403	. 2 ⊢ ℕ0 = (ℕ ∪ {0})
3	1, 2	sseqtrri 3262	1 ⊢ ℕ ⊆ ℕ0

Syntax hints:   ∪ cun 3198   ⊆ wss 3200  {csn 3669  0cc0 8032  ℕcn 9143  ℕ₀cn0 9402

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-n0 9403

This theorem is referenced by:  nnnn0  9409  nnnn0d  9455  expcnv  12070  oddge22np1  12447  bitsfzolem  12520