Theorem prmssnn 16612

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 16610	. 2 ⊢ (𝑥 ∈ ℙ → 𝑥 ∈ ℕ)
2	1	ssriv 3986	1 ⊢ ℙ ⊆ ℕ

Syntax hints:   ⊆ wss 3948  ℕcn 12211  ℙcprime 16607

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-prm 16608

This theorem is referenced by:  prmex  16613  prminf  16847  prmgaplem3  16985  prmgaplem4  16986  prmdvdsfi  26608  mumul  26682  sqff1o  26683  dirith2  27028  hgt750lema  33664  tgoldbachgtde  33667  tgoldbachgtda  33668  tgoldbachgt  33670  prmdvdsfmtnof1lem1  46242  prmdvdsfmtnof  46244