Theorem prmssnn 16645

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 16643	. 2 ⊢ (𝑥 ∈ ℙ → 𝑥 ∈ ℕ)
2	1	ssriv 3925	1 ⊢ ℙ ⊆ ℕ

Syntax hints:   ⊆ wss 3889  ℕcn 12174  ℙcprime 16640

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-prm 16641

This theorem is referenced by:  prmex  16646  prminf  16886  prmgaplem3  17024  prmgaplem4  17025  prmdvdsfi  27070  mumul  27144  sqff1o  27145  dirith2  27491  hgt750lema  34801  tgoldbachgtde  34804  tgoldbachgtda  34805  tgoldbachgt  34807  prmdvdsfmtnof1lem1  48047  prmdvdsfmtnof  48049