Theorem prmssnn 16615

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 16613	. 2 ⊢ (𝑥 ∈ ℙ → 𝑥 ∈ ℕ)
2	1	ssriv 3939	1 ⊢ ℙ ⊆ ℕ

Syntax hints:   ⊆ wss 3903  ℕcn 12157  ℙcprime 16610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-prm 16611

This theorem is referenced by:  prmex  16616  prminf  16855  prmgaplem3  16993  prmgaplem4  16994  prmdvdsfi  27085  mumul  27159  sqff1o  27160  dirith2  27507  hgt750lema  34834  tgoldbachgtde  34837  tgoldbachgtda  34838  tgoldbachgt  34840  prmdvdsfmtnof1lem1  47938  prmdvdsfmtnof  47940