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Theorem nnssre 8110
Description: The positive integers are a subset of the reals. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.)
Assertion
Ref Expression
nnssre  |-  NN  C_  RR

Proof of Theorem nnssre
StepHypRef Expression
1 1re 7180 . 2  |-  1  e.  RR
2 peano2re 7311 . . 3  |-  ( x  e.  RR  ->  (
x  +  1 )  e.  RR )
32rgen 2417 . 2  |-  A. x  e.  RR  ( x  + 
1 )  e.  RR
4 peano5nni 8109 . 2  |-  ( ( 1  e.  RR  /\  A. x  e.  RR  (
x  +  1 )  e.  RR )  ->  NN  C_  RR )
51, 3, 4mp2an 417 1  |-  NN  C_  RR
Colors of variables: wff set class
Syntax hints:    e. wcel 1434   A.wral 2349    C_ wss 2974  (class class class)co 5543   RRcr 7042   1c1 7044    + caddc 7046   NNcn 8106
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-cnex 7129  ax-resscn 7130  ax-1re 7132  ax-addrcl 7135
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-v 2604  df-in 2980  df-ss 2987  df-int 3645  df-inn 8107
This theorem is referenced by:  nnsscn  8111  nnre  8113  nnred  8119  nn0ssre  8359
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