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Theorem isprm 14703
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 4399 . . . 4  |-  ( p  =  P  ->  (
n  ||  p  <->  n  ||  P
) )
21rabbidv 3022 . . 3  |-  ( p  =  P  ->  { n  e.  NN  |  n  ||  p }  =  {
n  e.  NN  |  n  ||  P } )
32breq1d 4405 . 2  |-  ( p  =  P  ->  ( { n  e.  NN  |  n  ||  p }  ~~  2o  <->  { n  e.  NN  |  n  ||  P }  ~~  2o ) )
4 df-prm 14702 . 2  |-  Prime  =  { p  e.  NN  |  { n  e.  NN  |  n  ||  p }  ~~  2o }
53, 4elrab2 3186 1  |-  ( P  e.  Prime  <->  ( P  e.  NN  /\  { n  e.  NN  |  n  ||  P }  ~~  2o ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   {crab 2760   class class class wbr 4395   2oc2o 7194    ~~ cen 7584   NNcn 10631    || cdvds 14382   Primecprime 14701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-prm 14702
This theorem is referenced by:  prmnn  14704  1nprm  14708  isprm2  14711
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