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

Theorem isprm 10997
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 3826 . . . 4  |-  ( p  =  P  ->  (
n  ||  p  <->  n  ||  P
) )
21rabbidv 2604 . . 3  |-  ( p  =  P  ->  { n  e.  NN  |  n  ||  p }  =  {
n  e.  NN  |  n  ||  P } )
32breq1d 3832 . 2  |-  ( p  =  P  ->  ( { n  e.  NN  |  n  ||  p }  ~~  2o  <->  { n  e.  NN  |  n  ||  P }  ~~  2o ) )
4 df-prm 10996 . 2  |-  Prime  =  { p  e.  NN  |  { n  e.  NN  |  n  ||  p }  ~~  2o }
53, 4elrab2 2765 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 1287    e. wcel 1436   {crab 2359   class class class wbr 3822   2oc2o 6131    ~~ cen 6409   NNcn 8360    || cdvds 10702   Primecprime 10995
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rab 2364  df-v 2617  df-un 2992  df-sn 3437  df-pr 3438  df-op 3440  df-br 3823  df-prm 10996
This theorem is referenced by:  prmnn  10998  1nprm  11002  isprm2  11005
  Copyright terms: Public domain W3C validator