MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nnssnn0 Unicode version

Theorem nnssnn0 9984
Description: Positive naturals are a subset of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
Assertion
Ref Expression
nnssnn0  |-  NN  C_  NN0

Proof of Theorem nnssnn0
StepHypRef Expression
1 ssun1 3351 . 2  |-  NN  C_  ( NN  u.  { 0 } )
2 df-n0 9982 . 2  |-  NN0  =  ( NN  u.  { 0 } )
31, 2sseqtr4i 3224 1  |-  NN  C_  NN0
Colors of variables: wff set class
Syntax hints:    u. cun 3163    C_ wss 3165   {csn 3653   0cc0 8753   NNcn 9762   NN0cn0 9981
This theorem is referenced by:  nnnn0  9988  nnnn0d  10034  nthruz  12546  bitsfzolem  12641  ramub1  13091  ramcl  13092  ply1divex  19538  pserdvlem2  19820  hbtlem5  27435
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-un 3170  df-in 3172  df-ss 3179  df-n0 9982
  Copyright terms: Public domain W3C validator