Theorem prmssnn 16713

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 16711	. 2 ⊢ (𝑥 ∈ ℙ → 𝑥 ∈ ℕ)
2	1	ssriv 3987	1 ⊢ ℙ ⊆ ℕ

Syntax hints:   ⊆ wss 3951  ℕcn 12266  ℙcprime 16708

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-prm 16709

This theorem is referenced by:  prmex  16714  prminf  16953  prmgaplem3  17091  prmgaplem4  17092  prmdvdsfi  27150  mumul  27224  sqff1o  27225  dirith2  27572  hgt750lema  34672  tgoldbachgtde  34675  tgoldbachgtda  34676  tgoldbachgt  34678  prmdvdsfmtnof1lem1  47571  prmdvdsfmtnof  47573