Theorem nnssnn0 9138

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 3290	. 2 ⊢ ℕ ⊆ (ℕ ∪ {0})
2		df-n0 9136	. 2 ⊢ ℕ0 = (ℕ ∪ {0})
3	1, 2	sseqtrri 3182	1 ⊢ ℕ ⊆ ℕ0

Syntax hints:   ∪ cun 3119   ⊆ wss 3121  {csn 3583  0cc0 7774  ℕcn 8878  ℕ₀cn0 9135

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-n0 9136

This theorem is referenced by:  nnnn0  9142  nnnn0d  9188  expcnv  11467  oddge22np1  11840