Theorem prmssnn 16392

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 16390	. 2 ⊢ (𝑥 ∈ ℙ → 𝑥 ∈ ℕ)
2	1	ssriv 3930	1 ⊢ ℙ ⊆ ℕ

Syntax hints:   ⊆ wss 3892  ℕcn 11984  ℙcprime 16387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-br 5080  df-prm 16388

This theorem is referenced by:  prmex  16393  prminf  16627  prmgaplem3  16765  prmgaplem4  16766  prmdvdsfi  26267  mumul  26341  sqff1o  26342  dirith2  26687  hgt750lema  32646  tgoldbachgtde  32649  tgoldbachgtda  32650  tgoldbachgt  32652  prmdvdsfmtnof1lem1  45015  prmdvdsfmtnof  45017