Theorem prmssnn 16646

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 16644	. 2 ⊢ (𝑥 ∈ ℙ → 𝑥 ∈ ℕ)
2	1	ssriv 3950	1 ⊢ ℙ ⊆ ℕ

Syntax hints:   ⊆ wss 3914  ℕcn 12186  ℙcprime 16641

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 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-prm 16642

This theorem is referenced by:  prmex  16647  prminf  16886  prmgaplem3  17024  prmgaplem4  17025  prmdvdsfi  27017  mumul  27091  sqff1o  27092  dirith2  27439  hgt750lema  34648  tgoldbachgtde  34651  tgoldbachgtda  34652  tgoldbachgt  34654  prmdvdsfmtnof1lem1  47585  prmdvdsfmtnof  47587