Theorem prmssnn 16584

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

Step	Hyp	Ref	Expression
1		prmnn 16582	. 2 ⊢ (𝑥 ∈ ℙ → 𝑥 ∈ ℕ)
2	1	ssriv 3938	1 ⊢ ℙ ⊆ ℕ

Syntax hints:   ⊆ wss 3902  ℕcn 12122  ℙcprime 16579

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-prm 16580

This theorem is referenced by:  prmex  16585  prminf  16824  prmgaplem3  16962  prmgaplem4  16963  prmdvdsfi  27042  mumul  27116  sqff1o  27117  dirith2  27464  hgt750lema  34665  tgoldbachgtde  34668  tgoldbachgtda  34669  tgoldbachgt  34671  prmdvdsfmtnof1lem1  47614  prmdvdsfmtnof  47616