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Theorem isprm 13073
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 4208 . . . 4  |-  ( p  =  P  ->  (
n  ||  p  <->  n  ||  P
) )
21rabbidv 2940 . . 3  |-  ( p  =  P  ->  { n  e.  NN  |  n  ||  p }  =  {
n  e.  NN  |  n  ||  P } )
32breq1d 4214 . 2  |-  ( p  =  P  ->  ( { n  e.  NN  |  n  ||  p }  ~~  2o  <->  { n  e.  NN  |  n  ||  P }  ~~  2o ) )
4 df-prm 13072 . 2  |-  Prime  =  { p  e.  NN  |  { n  e.  NN  |  n  ||  p }  ~~  2o }
53, 4elrab2 3086 1  |-  ( P  e.  Prime  <->  ( P  e.  NN  /\  { n  e.  NN  |  n  ||  P }  ~~  2o ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   {crab 2701   class class class wbr 4204   2oc2o 6710    ~~ cen 7098   NNcn 9992    || cdivides 12844   Primecprime 13071
This theorem is referenced by:  prmnn  13074  1nprm  13076  isprm2  13079
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-prm 13072
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