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Theorem nn0ssre 9099
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 9096 . 2  |-  NN0  =  ( NN  u.  { 0 } )
2 nnssre 8842 . . 3  |-  NN  C_  RR
3 0re 7880 . . . 4  |-  0  e.  RR
4 snssi 3702 . . . 4  |-  ( 0  e.  RR  ->  { 0 }  C_  RR )
53, 4ax-mp 5 . . 3  |-  { 0 }  C_  RR
62, 5unssi 3283 . 2  |-  ( NN  u.  { 0 } )  C_  RR
71, 6eqsstri 3160 1  |-  NN0  C_  RR
Colors of variables: wff set class
Syntax hints:    e. wcel 2128    u. cun 3100    C_ wss 3102   {csn 3561   RRcr 7733   0cc0 7734   NNcn 8838   NN0cn0 9095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139  ax-sep 4084  ax-cnex 7825  ax-resscn 7826  ax-1re 7828  ax-addrcl 7831  ax-rnegex 7843
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-sn 3567  df-int 3810  df-inn 8839  df-n0 9096
This theorem is referenced by:  nn0sscn  9100  nn0re  9104  nn0rei  9106  nn0red  9149
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