Theorem nnssnn0 11172

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 3738	. 2 ⊢ ℕ ⊆ (ℕ ∪ {0})
2		df-n0 11170	. 2 ⊢ ℕ0 = (ℕ ∪ {0})
3	1, 2	sseqtr4i 3601	1 ⊢ ℕ ⊆ ℕ0

Syntax hints:   ∪ cun 3538   ⊆ wss 3540  {csn 4125  0cc0 9815  ℕcn 10897  ℕ₀cn0 11169

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-un 3545  df-in 3547  df-ss 3554  df-n0 11170

This theorem is referenced by:  nnnn0  11176  nnnn0d  11228  nthruz  14820  oddge22np1  14911  bitsfzolem  14994  lcmfval  15172  ramub1  15570  ramcl  15571  ply1divex  23700  pserdvlem2  23986  knoppndvlem18  31690  hbtlem5  36717  brfvtrcld  37032  corcltrcl  37050  fourierdlem50  39049  fourierdlem102  39101  fourierdlem114  39113  fmtnoinf  39986  fmtnofac2  40019