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Mirrors > Home > MPE Home > Th. List > prmssnn | Structured version Visualization version GIF version |
Description: The prime numbers are a subset of the positive integers. (Contributed by AV, 22-Jul-2020.) |
Ref | Expression |
---|---|
prmssnn | ⊢ ℙ ⊆ ℕ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prmnn 16648 | . 2 ⊢ (𝑥 ∈ ℙ → 𝑥 ∈ ℕ) | |
2 | 1 | ssriv 3980 | 1 ⊢ ℙ ⊆ ℕ |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3944 ℕcn 12245 ℙcprime 16645 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5150 df-prm 16646 |
This theorem is referenced by: prmex 16651 prminf 16887 prmgaplem3 17025 prmgaplem4 17026 prmdvdsfi 27084 mumul 27158 sqff1o 27159 dirith2 27506 hgt750lema 34420 tgoldbachgtde 34423 tgoldbachgtda 34424 tgoldbachgt 34426 prmdvdsfmtnof1lem1 47061 prmdvdsfmtnof 47063 |
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