Theorem prmssnn 16724

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 16722	. 2 ⊢ (𝑥 ∈ ℙ → 𝑥 ∈ ℕ)
2	1	ssriv 3943	1 ⊢ ℙ ⊆ ℕ

Syntax hints:   ⊆ wss 3907  ℕcn 12224  ℙcprime 16719

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-prm 16720

This theorem is referenced by:  prmex  16725  prminf  16965  prmgaplem3  17103  prmgaplem4  17104  prmdvdsfi  27229  mumul  27303  sqff1o  27304  dirith2  27650  hgt750lema  34961  tgoldbachgtde  34964  tgoldbachgtda  34965  tgoldbachgt  34967  prmdvdsfmtnof1lem1  48191  prmdvdsfmtnof  48193