Theorem nnssnn0 12531

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 4124	. 2 ⊢ ℕ ⊆ (ℕ ∪ {0})
2		df-n0 12529	. 2 ⊢ ℕ0 = (ℕ ∪ {0})
3	1, 2	sseqtrri 3980	1 ⊢ ℕ ⊆ ℕ0

This proof depends on syntax axioms:   ∪ cun 3897   ⊆ wss 3899  {csn 4584  0cc0 11124  ℕcn 12257  ℕ₀cn0 12528

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-un 3904  df-ss 3916  df-n0 12529

This theorem is used by:  nnnn0  12535  nnnn0d  12589  nthruz  16341  oddge22np1  16439  bitsfzolem  16524  lcmfval  16711  ramub1  17120  ramcl  17121  ply1divex  26362  pserdvlem2  26664  2sqreunnlem1  27685  2sqreunnlem2  27691  fsum2dsub  35115  breprexplemc  35140  breprexpnat  35142  knoppndvlem18  37226  sumcubes  43188  hbtlem5  43969  brfvtrcld  44561  corcltrcl  44579  fourierdlem50  46984  fourierdlem102  47036  fourierdlem114  47048  fmtnoinf  48439  fmtnofac2  48472