MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  prmssnn Structured version   Visualization version   GIF version

Theorem prmssnn 15371
Description: The prime numbers are a subset of the positive integers. (Contributed by AV, 22-Jul-2020.)
Assertion
Ref Expression
prmssnn ℙ ⊆ ℕ

Proof of Theorem prmssnn
StepHypRef Expression
1 prmnn 15369 . 2 (𝑥 ∈ ℙ → 𝑥 ∈ ℕ)
21ssriv 3599 1 ℙ ⊆ ℕ
Colors of variables: wff setvar class
Syntax hints:  wss 3567  cn 11005  cprime 15366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-br 4645  df-prm 15367
This theorem is referenced by:  prmex  15372  prmgaplem3  15738  prmgaplem4  15739  hgt750lema  30709  tgoldbachgtde  30712  tgoldbachgtda  30713  tgoldbachgt  30715  prmdvdsfmtnof1lem1  41261  prmdvdsfmtnof  41263
  Copyright terms: Public domain W3C validator