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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 11717 | . 2 ⊢ (𝑃 ∈ ℙ ↔ (𝑃 ∈ ℕ ∧ {𝑧 ∈ ℕ ∣ 𝑧 ∥ 𝑃} ≈ 2o)) | |
2 | 1 | simplbi 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 ℕcn 8688 ∥ cdvds 11420 ℙcprime 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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