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Mirrors > Home > ILE Home > Th. List > prmnn | GIF version |
Description: A prime number is a positive integer. (Contributed by Paul Chapman, 22-Jun-2011.) |
Ref | Expression |
---|---|
prmnn | ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isprm 10491 | . 2 ⊢ (𝑃 ∈ ℙ ↔ (𝑃 ∈ ℕ ∧ {𝑧 ∈ ℕ ∣ 𝑧 ∥ 𝑃} ≈ 2𝑜)) | |
2 | 1 | simplbi 268 | 1 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1433 {crab 2352 class class class wbr 3785 2𝑜c2o 6018 ≈ cen 6242 ℕcn 8039 ∥ cdvds 10195 ℙcprime 10489 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rab 2357 df-v 2603 df-un 2977 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-prm 10490 |
This theorem is referenced by: prmz 10493 prmssnn 10494 nprmdvds1 10521 coprm 10523 euclemma 10525 prmdvdsexpr 10529 cncongrprm 10536 |
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