Theorem isprm 11790

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 3933	. . . 4 |- ( p = P -> ( n || p <-> n || P ) )
2	1	rabbidv 2675	. . 3 |- ( p = P -> { n e. NN | n || p } = { n e. NN | n || P } )
3	2	breq1d 3939	. 2 |- ( p = P -> ( { n e. NN | n || p } ~~ 2o <-> { n e. NN | n || P } ~~ 2o ) )
4		df-prm 11789	. 2 |- Prime = { p e. NN | { n e. NN | n || p } ~~ 2o }
5	3, 4	elrab2 2843	1 |- ( P e. Prime <-> ( P e. NN /\ { n e. NN | n || P } ~~ 2o ) )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   {crab 2420   class class class wbr 3929   2oc2o 6307   ~~ cen 6632   NNcn 8720   || cdvds 11493   Primecprime 11788

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rab 2425  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-prm 11789

This theorem is referenced by:  prmnn  11791  1nprm  11795  isprm2  11798