Theorem isprm 13768

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

Step	Hyp	Ref	Expression
1		breq2 3342	. . . . 5 |- (p = P -> (n||p <-> n||P))
2	1	adantr 425	. . . 4 |- ((p = P /\ n e. NN) -> (n||p <-> n||P))
3	2	rabbidva 2286	. . 3 |- (p = P -> {n e. NN | n||p} = {n e. NN | n||P})
4	3	breq1d 3348	. 2 |- (p = P -> ({n e. NN | n||p} ~~ 2o <-> {n e. NN | n||P} ~~ 2o))
5		df-prime 13767	. 2 |- Prime = {p e. NN | {n e. NN | n||p} ~~ 2o}
6	4, 5	elrab2 2416	1 |- (P e. Prime <-> (P e. NN /\ {n e. NN | n||P} ~~ 2o))

Syntax hints:   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  {crab 2108   class class class wbr 3338  2oc2o 5173   ~~ cen 5423  NNcn 6449  ||cdivides 13662  Primecprime 13766

This theorem is referenced by:  1nprm 13769  isprm2 13775  prmnn 13778

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-rab 2112  df-v 2294  df-un 2600  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-prime 13767

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