Theorem prmssnn 16014

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 16012	. 2 ⊢ (𝑥 ∈ ℙ → 𝑥 ∈ ℕ)
2	1	ssriv 3971	1 ⊢ ℙ ⊆ ℕ

Syntax hints:   ⊆ wss 3936  ℕcn 11632  ℙcprime 16009

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3497  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-br 5060  df-prm 16010

This theorem is referenced by:  prmex  16015  prminf  16245  prmgaplem3  16383  prmgaplem4  16384  prmdvdsfi  25678  mumul  25752  sqff1o  25753  dirith2  26098  hgt750lema  31923  tgoldbachgtde  31926  tgoldbachgtda  31927  tgoldbachgt  31929  prmdvdsfmtnof1lem1  43739  prmdvdsfmtnof  43741