Theorem isprm 12546

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 4063	. . . 4 |- ( p = P -> ( n || p <-> n || P ) )
2	1	rabbidv 2765	. . 3 |- ( p = P -> { n e. NN | n || p } = { n e. NN | n || P } )
3	2	breq1d 4069	. 2 |- ( p = P -> ( { n e. NN | n || p } ~~ 2o <-> { n e. NN | n || P } ~~ 2o ) )
4		df-prm 12545	. 2 |- Prime = { p e. NN | { n e. NN | n || p } ~~ 2o }
5	3, 4	elrab2 2939	1 |- ( P e. Prime <-> ( P e. NN /\ { n e. NN | n || P } ~~ 2o ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2178   {crab 2490   class class class wbr 4059   2oc2o 6519   ~~ cen 6848   NNcn 9071   || cdvds 12213   Primecprime 12544

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rab 2495  df-v 2778  df-un 3178  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-prm 12545

This theorem is referenced by:  prmnn  12547  1nprm  12551  isprm2  12554