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Theorem prmnn 11798
Description: A prime number is a positive integer. (Contributed by Paul Chapman, 22-Jun-2011.)
Assertion
Ref Expression
prmnn  |-  ( P  e.  Prime  ->  P  e.  NN )

Proof of Theorem prmnn
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 isprm 11797 . 2  |-  ( P  e.  Prime  <->  ( P  e.  NN  /\  { z  e.  NN  |  z 
||  P }  ~~  2o ) )
21simplbi 272 1  |-  ( P  e.  Prime  ->  P  e.  NN )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1480   {crab 2420   class class class wbr 3929   2oc2o 6307    ~~ cen 6632   NNcn 8727    || cdvds 11500   Primecprime 11795
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rab 2425  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-prm 11796
This theorem is referenced by:  prmz  11799  prmssnn  11800  nprmdvds1  11827  coprm  11829  euclemma  11831  prmdvdsexpr  11835  cncongrprm  11842  phiprmpw  11905
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