Theorem prmssnn 16390

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 16388	. 2 ⊢ (𝑥 ∈ ℙ → 𝑥 ∈ ℕ)
2	1	ssriv 3926	1 ⊢ ℙ ⊆ ℕ

Syntax hints:   ⊆ wss 3888  ℕcn 11982  ℙcprime 16385

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2710

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2069  df-clab 2717  df-cleq 2731  df-clel 2817  df-rab 3074  df-v 3435  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4258  df-if 4461  df-sn 4563  df-pr 4565  df-op 4569  df-br 5076  df-prm 16386

This theorem is referenced by:  prmex  16391  prminf  16625  prmgaplem3  16763  prmgaplem4  16764  prmdvdsfi  26265  mumul  26339  sqff1o  26340  dirith2  26685  hgt750lema  32646  tgoldbachgtde  32649  tgoldbachgtda  32650  tgoldbachgt  32652  prmdvdsfmtnof1lem1  45047  prmdvdsfmtnof  45049