Theorem nnssnn0 12529

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 4178	. 2 ⊢ ℕ ⊆ (ℕ ∪ {0})
2		df-n0 12527	. 2 ⊢ ℕ0 = (ℕ ∪ {0})
3	1, 2	sseqtrri 4033	1 ⊢ ℕ ⊆ ℕ0

Syntax hints:   ∪ cun 3949   ⊆ wss 3951  {csn 4626  0cc0 11155  ℕcn 12266  ℕ₀cn0 12526

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-un 3956  df-ss 3968  df-n0 12527

This theorem is referenced by:  nnnn0  12533  nnnn0d  12587  nthruz  16289  oddge22np1  16386  bitsfzolem  16471  lcmfval  16658  ramub1  17066  ramcl  17067  ply1divex  26176  pserdvlem2  26472  2sqreunnlem1  27493  2sqreunnlem2  27499  fsum2dsub  34622  breprexplemc  34647  breprexpnat  34649  knoppndvlem18  36530  sumcubes  42347  hbtlem5  43140  brfvtrcld  43734  corcltrcl  43752  fourierdlem50  46171  fourierdlem102  46223  fourierdlem114  46235  fmtnoinf  47523  fmtnofac2  47556