Theorem prmssnn 16010

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 16008	. 2 ⊢ (𝑥 ∈ ℙ → 𝑥 ∈ ℕ)
2	1	ssriv 3919	1 ⊢ ℙ ⊆ ℕ

Syntax hints:   ⊆ wss 3881  ℕcn 11625  ℙcprime 16005

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-rab 3115  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-prm 16006

This theorem is referenced by:  prmex  16011  prminf  16241  prmgaplem3  16379  prmgaplem4  16380  prmdvdsfi  25692  mumul  25766  sqff1o  25767  dirith2  26112  hgt750lema  32038  tgoldbachgtde  32041  tgoldbachgtda  32042  tgoldbachgt  32044  prmdvdsfmtnof1lem1  44101  prmdvdsfmtnof  44103