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Theorem nn0ssre 8738
Description: Nonnegative integers are a subset of the reals. (Contributed by Raph Levien, 10-Dec-2002.)
Assertion
Ref Expression
nn0ssre  |-  NN0  C_  RR

Proof of Theorem nn0ssre
StepHypRef Expression
1 df-n0 8735 . 2  |-  NN0  =  ( NN  u.  { 0 } )
2 nnssre 8487 . . 3  |-  NN  C_  RR
3 0re 7549 . . . 4  |-  0  e.  RR
4 snssi 3587 . . . 4  |-  ( 0  e.  RR  ->  { 0 }  C_  RR )
53, 4ax-mp 7 . . 3  |-  { 0 }  C_  RR
62, 5unssi 3176 . 2  |-  ( NN  u.  { 0 } )  C_  RR
71, 6eqsstri 3057 1  |-  NN0  C_  RR
Colors of variables: wff set class
Syntax hints:    e. wcel 1439    u. cun 2998    C_ wss 3000   {csn 3450   RRcr 7410   0cc0 7411   NNcn 8483   NN0cn0 8734
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-cnex 7497  ax-resscn 7498  ax-1re 7500  ax-addrcl 7503  ax-rnegex 7515
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-sn 3456  df-int 3695  df-inn 8484  df-n0 8735
This theorem is referenced by:  nn0sscn  8739  nn0re  8743  nn0rei  8745  nn0red  8788
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