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Theorem nn0ssre 8243
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 8240 . 2  |-  NN0  =  ( NN  u.  { 0 } )
2 nnssre 7994 . . 3  |-  NN  C_  RR
3 0re 7085 . . . 4  |-  0  e.  RR
4 snssi 3536 . . . 4  |-  ( 0  e.  RR  ->  { 0 }  C_  RR )
53, 4ax-mp 7 . . 3  |-  { 0 }  C_  RR
62, 5unssi 3146 . 2  |-  ( NN  u.  { 0 } )  C_  RR
71, 6eqsstri 3003 1  |-  NN0  C_  RR
Colors of variables: wff set class
Syntax hints:    e. wcel 1409    u. cun 2943    C_ wss 2945   {csn 3403   RRcr 6946   0cc0 6947   NNcn 7990   NN0cn0 8239
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-cnex 7033  ax-resscn 7034  ax-1re 7036  ax-addrcl 7039  ax-rnegex 7051
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-sn 3409  df-int 3644  df-inn 7991  df-n0 8240
This theorem is referenced by:  nn0sscn  8244  nn0re  8248  nn0rei  8250  nn0red  8293
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