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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 15767 | . 2 ⊢ (𝑥 ∈ ℙ → 𝑥 ∈ ℕ) | |
2 | 1 | ssriv 3831 | 1 ⊢ ℙ ⊆ ℕ |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3798 ℕcn 11357 ℙcprime 15764 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-br 4876 df-prm 15765 |
This theorem is referenced by: prmex 15770 prmgaplem3 16135 prmgaplem4 16136 hgt750lema 31280 tgoldbachgtde 31283 tgoldbachgtda 31284 tgoldbachgt 31286 prmdvdsfmtnof1lem1 42340 prmdvdsfmtnof 42342 |
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