Theorem prmssnn 16022

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 16020	. 2 ⊢ (𝑥 ∈ ℙ → 𝑥 ∈ ℕ)
2	1	ssriv 3973	1 ⊢ ℙ ⊆ ℕ

Syntax hints:   ⊆ wss 3938  ℕcn 11640  ℙcprime 16017

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-br 5069  df-prm 16018

This theorem is referenced by:  prmex  16023  prminf  16253  prmgaplem3  16391  prmgaplem4  16392  prmdvdsfi  25686  mumul  25760  sqff1o  25761  dirith2  26106  hgt750lema  31930  tgoldbachgtde  31933  tgoldbachgtda  31934  tgoldbachgt  31936  prmdvdsfmtnof1lem1  43753  prmdvdsfmtnof  43755