Theorem prmssnn 16605

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 16603	. 2 ⊢ (𝑥 ∈ ℙ → 𝑥 ∈ ℕ)
2	1	ssriv 3941	1 ⊢ ℙ ⊆ ℕ

Syntax hints:   ⊆ wss 3905  ℕcn 12146  ℙcprime 16600

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 3397  df-v 3440  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-prm 16601

This theorem is referenced by:  prmex  16606  prminf  16845  prmgaplem3  16983  prmgaplem4  16984  prmdvdsfi  27033  mumul  27107  sqff1o  27108  dirith2  27455  hgt750lema  34624  tgoldbachgtde  34627  tgoldbachgtda  34628  tgoldbachgt  34630  prmdvdsfmtnof1lem1  47569  prmdvdsfmtnof  47571