Theorem nnssnn0 12440

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 4118	. 2 ⊢ ℕ ⊆ (ℕ ∪ {0})
2		df-n0 12438	. 2 ⊢ ℕ0 = (ℕ ∪ {0})
3	1, 2	sseqtrri 3971	1 ⊢ ℕ ⊆ ℕ0

Syntax hints:   ∪ cun 3887   ⊆ wss 3889  {csn 4567  0cc0 11038  ℕcn 12174  ℕ₀cn0 12437

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3431  df-un 3894  df-ss 3906  df-n0 12438

This theorem is referenced by:  nnnn0  12444  nnnn0d  12498  nthruz  16220  oddge22np1  16318  bitsfzolem  16403  lcmfval  16590  ramub1  16999  ramcl  17000  ply1divex  26102  pserdvlem2  26393  2sqreunnlem1  27412  2sqreunnlem2  27418  fsum2dsub  34751  breprexplemc  34776  breprexpnat  34778  knoppndvlem18  36789  sumcubes  42745  hbtlem5  43556  brfvtrcld  44148  corcltrcl  44166  fourierdlem50  46584  fourierdlem102  46636  fourierdlem114  46648  fmtnoinf  47999  fmtnofac2  48032