Theorem prmssnn 16654

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 16652	. 2 ⊢ (𝑥 ∈ ℙ → 𝑥 ∈ ℕ)
2	1	ssriv 3986	1 ⊢ ℙ ⊆ ℕ

Syntax hints:   ⊆ wss 3949  ℕcn 12250  ℙcprime 16649

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 2699

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-prm 16650

This theorem is referenced by:  prmex  16655  prminf  16891  prmgaplem3  17029  prmgaplem4  17030  prmdvdsfi  27059  mumul  27133  sqff1o  27134  dirith2  27481  hgt750lema  34322  tgoldbachgtde  34325  tgoldbachgtda  34326  tgoldbachgt  34328  prmdvdsfmtnof1lem1  46953  prmdvdsfmtnof  46955