Theorem prmnn 13778

Description: A prime number is a positive integer. (Contributed by Paul Chapman, 22-Jun-2011.)

Assertion

Ref	Expression
prmnn	|- (P e. Prime -> P e. NN)

Proof of Theorem prmnn

Step	Hyp	Ref	Expression
1		isprm 13768	. . 3 |- (P e. Prime <-> (P e. NN /\ {z e. NN | z||P} ~~ 2o))
2	1	biimpi 168	. 2 |- (P e. Prime -> (P e. NN /\ {z e. NN | z||P} ~~ 2o))
3	2	simplld 348	1 |- (P e. Prime -> P e. NN)

Syntax hints:   -> wi 3   /\ wa 240   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:  coprm 13782

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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