Theorem prmssnn 16618

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 16616	. 2 ⊢ (𝑥 ∈ ℙ → 𝑥 ∈ ℕ)
2	1	ssriv 3981	1 ⊢ ℙ ⊆ ℕ

Syntax hints:   ⊆ wss 3943  ℕcn 12213  ℙcprime 16613

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-prm 16614

This theorem is referenced by:  prmex  16619  prminf  16855  prmgaplem3  16993  prmgaplem4  16994  prmdvdsfi  26990  mumul  27064  sqff1o  27065  dirith2  27412  hgt750lema  34198  tgoldbachgtde  34201  tgoldbachgtda  34202  tgoldbachgt  34204  prmdvdsfmtnof1lem1  46805  prmdvdsfmtnof  46807