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| 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 16711 | . 2 ⊢ (𝑥 ∈ ℙ → 𝑥 ∈ ℕ) | |
| 2 | 1 | ssriv 3987 | 1 ⊢ ℙ ⊆ ℕ | 
| Colors of variables: wff setvar class | 
| Syntax hints: ⊆ wss 3951 ℕcn 12266 ℙcprime 16708 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-prm 16709 | 
| This theorem is referenced by: prmex 16714 prminf 16953 prmgaplem3 17091 prmgaplem4 17092 prmdvdsfi 27150 mumul 27224 sqff1o 27225 dirith2 27572 hgt750lema 34672 tgoldbachgtde 34675 tgoldbachgtda 34676 tgoldbachgt 34678 prmdvdsfmtnof1lem1 47571 prmdvdsfmtnof 47573 | 
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