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Theorem prmnn 11718
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 11717 . 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 1465  {crab 2397   class class class wbr 3899  2oc2o 6275  cen 6600  cn 8688  cdvds 11420  cprime 11715
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rab 2402  df-v 2662  df-un 3045  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-prm 11716
This theorem is referenced by:  prmz  11719  prmssnn  11720  nprmdvds1  11747  coprm  11749  euclemma  11751  prmdvdsexpr  11755  cncongrprm  11762  phiprmpw  11825
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