Theorem prmssnn 16589

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 16587	. 2 ⊢ (𝑥 ∈ ℙ → 𝑥 ∈ ℕ)
2	1	ssriv 3934	1 ⊢ ℙ ⊆ ℕ

Syntax hints:   ⊆ wss 3898  ℕcn 12132  ℙcprime 16584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-br 5094  df-prm 16585

This theorem is referenced by:  prmex  16590  prminf  16829  prmgaplem3  16967  prmgaplem4  16968  prmdvdsfi  27045  mumul  27119  sqff1o  27120  dirith2  27467  hgt750lema  34691  tgoldbachgtde  34694  tgoldbachgtda  34695  tgoldbachgt  34697  prmdvdsfmtnof1lem1  47708  prmdvdsfmtnof  47710