Theorem isprm 12063

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 3993	. . . 4 |- ( p = P -> ( n || p <-> n || P ) )
2	1	rabbidv 2719	. . 3 |- ( p = P -> { n e. NN | n || p } = { n e. NN | n || P } )
3	2	breq1d 3999	. 2 |- ( p = P -> ( { n e. NN | n || p } ~~ 2o <-> { n e. NN | n || P } ~~ 2o ) )
4		df-prm 12062	. 2 |- Prime = { p e. NN | { n e. NN | n || p } ~~ 2o }
5	3, 4	elrab2 2889	1 |- ( P e. Prime <-> ( P e. NN /\ { n e. NN | n || P } ~~ 2o ) )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   {crab 2452   class class class wbr 3989   2oc2o 6389   ~~ cen 6716   NNcn 8878   || cdvds 11749   Primecprime 12061

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rab 2457  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-prm 12062

This theorem is referenced by:  prmnn  12064  1nprm  12068  isprm2  12071