Theorem prmssnn 16723

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 16721	. 2 ⊢ (𝑥 ∈ ℙ → 𝑥 ∈ ℕ)
2	1	ssriv 4012	1 ⊢ ℙ ⊆ ℕ

Syntax hints:   ⊆ wss 3976  ℕcn 12293  ℙcprime 16718

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-prm 16719

This theorem is referenced by:  prmex  16724  prminf  16962  prmgaplem3  17100  prmgaplem4  17101  prmdvdsfi  27168  mumul  27242  sqff1o  27243  dirith2  27590  hgt750lema  34634  tgoldbachgtde  34637  tgoldbachgtda  34638  tgoldbachgt  34640  prmdvdsfmtnof1lem1  47458  prmdvdsfmtnof  47460