ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  isprm Unicode version

Theorem isprm 10716
Description: The predicate "is a prime number". A prime number is a positive integer with exactly two positive divisors. (Contributed by Paul Chapman, 22-Jun-2011.)
Assertion
Ref Expression
isprm  |-  ( P  e.  Prime  <->  ( P  e.  NN  /\  { n  e.  NN  |  n  ||  P }  ~~  2o ) )
Distinct variable group:    P, n

Proof of Theorem isprm
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 breq2 3809 . . . 4  |-  ( p  =  P  ->  (
n  ||  p  <->  n  ||  P
) )
21rabbidv 2599 . . 3  |-  ( p  =  P  ->  { n  e.  NN  |  n  ||  p }  =  {
n  e.  NN  |  n  ||  P } )
32breq1d 3815 . 2  |-  ( p  =  P  ->  ( { n  e.  NN  |  n  ||  p }  ~~  2o  <->  { n  e.  NN  |  n  ||  P }  ~~  2o ) )
4 df-prm 10715 . 2  |-  Prime  =  { p  e.  NN  |  { n  e.  NN  |  n  ||  p }  ~~  2o }
53, 4elrab2 2760 1  |-  ( P  e.  Prime  <->  ( P  e.  NN  /\  { n  e.  NN  |  n  ||  P }  ~~  2o ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   {crab 2357   class class class wbr 3805   2oc2o 6080    ~~ cen 6307   NNcn 8176    || cdvds 10421   Primecprime 10714
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 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rab 2362  df-v 2612  df-un 2986  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-prm 10715
This theorem is referenced by:  prmnn  10717  1nprm  10721  isprm2  10724
  Copyright terms: Public domain W3C validator