Theorem prmssnn 16309

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 16307	. 2 ⊢ (𝑥 ∈ ℙ → 𝑥 ∈ ℕ)
2	1	ssriv 3921	1 ⊢ ℙ ⊆ ℕ

Syntax hints:   ⊆ wss 3883  ℕcn 11903  ℙcprime 16304

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-prm 16305

This theorem is referenced by:  prmex  16310  prminf  16544  prmgaplem3  16682  prmgaplem4  16683  prmdvdsfi  26161  mumul  26235  sqff1o  26236  dirith2  26581  hgt750lema  32537  tgoldbachgtde  32540  tgoldbachgtda  32541  tgoldbachgt  32543  prmdvdsfmtnof1lem1  44924  prmdvdsfmtnof  44926