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Theorem prmnn 11780
 Description: A prime number is a positive integer. (Contributed by Paul Chapman, 22-Jun-2011.)
Assertion
Ref Expression
prmnn (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)

Proof of Theorem prmnn
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 isprm 11779 . 2 (𝑃 ∈ ℙ ↔ (𝑃 ∈ ℕ ∧ {𝑧 ∈ ℕ ∣ 𝑧𝑃} ≈ 2o))
21simplbi 272 1 (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 1480  {crab 2418   class class class wbr 3924  2oc2o 6300   ≈ cen 6625  ℕcn 8713   ∥ cdvds 11482  ℙcprime 11777 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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rab 2423  df-v 2683  df-un 3070  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-prm 11778 This theorem is referenced by:  prmz  11781  prmssnn  11782  nprmdvds1  11809  coprm  11811  euclemma  11813  prmdvdsexpr  11817  cncongrprm  11824  phiprmpw  11887
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