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Theorem isprm 12776
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 4043 . . . 4  |-  ( p  =  P  ->  (
n  ||  p  <->  n  ||  P
) )
21rabbidv 2793 . . 3  |-  ( p  =  P  ->  { n  e.  NN  |  n  ||  p }  =  {
n  e.  NN  |  n  ||  P } )
32breq1d 4049 . 2  |-  ( p  =  P  ->  ( { n  e.  NN  |  n  ||  p }  ~~  2o  <->  { n  e.  NN  |  n  ||  P }  ~~  2o ) )
4 df-prm 12775 . 2  |-  Prime  =  { p  e.  NN  |  { n  e.  NN  |  n  ||  p }  ~~  2o }
53, 4elrab2 2938 1  |-  ( P  e.  Prime  <->  ( P  e.  NN  /\  { n  e.  NN  |  n  ||  P }  ~~  2o ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   {crab 2560   class class class wbr 4039   2oc2o 6489    ~~ cen 6876   NNcn 9762    || cdivides 12547   Primecprime 12774
This theorem is referenced by:  prmnn  12777  1nprm  12779  isprm2  12782
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-prm 12775
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