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Theorem nn0ssre 9139
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 9136 . 2  |-  NN0  =  ( NN  u.  { 0 } )
2 nnssre 8882 . . 3  |-  NN  C_  RR
3 0re 7920 . . . 4  |-  0  e.  RR
4 snssi 3724 . . . 4  |-  ( 0  e.  RR  ->  { 0 }  C_  RR )
53, 4ax-mp 5 . . 3  |-  { 0 }  C_  RR
62, 5unssi 3302 . 2  |-  ( NN  u.  { 0 } )  C_  RR
71, 6eqsstri 3179 1  |-  NN0  C_  RR
Colors of variables: wff set class
Syntax hints:    e. wcel 2141    u. cun 3119    C_ wss 3121   {csn 3583   RRcr 7773   0cc0 7774   NNcn 8878   NN0cn0 9135
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-sep 4107  ax-cnex 7865  ax-resscn 7866  ax-1re 7868  ax-addrcl 7871  ax-rnegex 7883
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-int 3832  df-inn 8879  df-n0 9136
This theorem is referenced by:  nn0sscn  9140  nn0re  9144  nn0rei  9146  nn0red  9189
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