Theorem isprm 12865

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 4129	. . . 4 |- ( p = P -> ( n || p <-> n || P ) )
2	1	rabbidv 2810	. . 3 |- ( p = P -> { n e. NN | n || p } = { n e. NN | n || P } )
3	2	breq1d 4135	. 2 |- ( p = P -> ( { n e. NN | n || p } ~~ 2o <-> { n e. NN | n || P } ~~ 2o ) )
4		df-prm 12864	. 2 |- Prime = { p e. NN | { n e. NN | n || p } ~~ 2o }
5	3, 4	elrab2 2985	1 |- ( P e. Prime <-> ( P e. NN /\ { n e. NN | n || P } ~~ 2o ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   {crab 2532   class class class wbr 4125   2oc2o 6671   ~~ cen 7010   NNcn 9283   || cdvds 12532   Primecprime 12863

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rab 2537  df-v 2823  df-un 3224  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-prm 12864

This theorem is referenced by:  prmnn  12866  1nprm  12870  isprm2  12873