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

Theorem nnssnn0 9968
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 3338 . 2  |-  NN  C_  ( NN  u.  { 0 } )
2 df-n0 9966 . 2  |-  NN0  =  ( NN  u.  { 0 } )
31, 2sseqtr4i 3211 1  |-  NN  C_  NN0
Colors of variables: wff set class
Syntax hints:    u. cun 3150    C_ wss 3152   {csn 3640   0cc0 8737   NNcn 9746   NN0cn0 9965
This theorem is referenced by:  nnnn0  9972  nnnn0d  10018  nthruz  12530  bitsfzolem  12625  ramub1  13075  ramcl  13076  ply1divex  19522  pserdvlem2  19804  hbtlem5  27332
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-un 3157  df-in 3159  df-ss 3166  df-n0 9966
  Copyright terms: Public domain W3C validator