Theorem isprm 10491

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.

Step	Hyp	Ref	Expression
1		breq2 3789	. . . 4 |- ( p = P -> ( n || p <-> n || P ) )
2	1	rabbidv 2593	. . 3 |- ( p = P -> { n e. NN | n || p } = { n e. NN | n || P } )
3	2	breq1d 3795	. 2 |- ( p = P -> ( { n e. NN | n || p } ~~ 2o <-> { n e. NN | n || P } ~~ 2o ) )
4		df-prm 10490	. 2 |- Prime = { p e. NN | { n e. NN | n || p } ~~ 2o }
5	3, 4	elrab2 2751	1 |- ( P e. Prime <-> ( P e. NN /\ { n e. NN | n || P } ~~ 2o ) )

Syntax hints:   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   {crab 2352   class class class wbr 3785   2oc2o 6018   ~~ cen 6242   NNcn 8039   || cdvds 10195   Primecprime 10489

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rab 2357  df-v 2603  df-un 2977  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-prm 10490

This theorem is referenced by:  prmnn  10492  1nprm  10496  isprm2  10499