Theorem prmssnn 16574

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 16572	. 2 ⊢ (𝑥 ∈ ℙ → 𝑥 ∈ ℕ)
2	1	ssriv 3935	1 ⊢ ℙ ⊆ ℕ

Syntax hints:   ⊆ wss 3899  ℕcn 12116  ℙcprime 16569

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3393  df-v 3435  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5089  df-prm 16570

This theorem is referenced by:  prmex  16575  prminf  16814  prmgaplem3  16952  prmgaplem4  16953  prmdvdsfi  26998  mumul  27072  sqff1o  27073  dirith2  27420  hgt750lema  34638  tgoldbachgtde  34641  tgoldbachgtda  34642  tgoldbachgt  34644  prmdvdsfmtnof1lem1  47582  prmdvdsfmtnof  47584