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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 11790 | . 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 1480 {crab 2420 class class class wbr 3929 2oc2o 6307 ≈ cen 6632 ℕcn 8720 ∥ cdvds 11493 ℙcprime 11788 |
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 2121 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rab 2425 df-v 2688 df-un 3075 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-prm 11789 |
This theorem is referenced by: prmz 11792 prmssnn 11793 nprmdvds1 11820 coprm 11822 euclemma 11824 prmdvdsexpr 11828 cncongrprm 11835 phiprmpw 11898 |
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