Theorem prmssnn 16653

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 16651	. 2 ⊢ (𝑥 ∈ ℙ → 𝑥 ∈ ℕ)
2	1	ssriv 3953	1 ⊢ ℙ ⊆ ℕ

Syntax hints:   ⊆ wss 3917  ℕcn 12193  ℙcprime 16648

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 2702

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 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-prm 16649

This theorem is referenced by:  prmex  16654  prminf  16893  prmgaplem3  17031  prmgaplem4  17032  prmdvdsfi  27024  mumul  27098  sqff1o  27099  dirith2  27446  hgt750lema  34655  tgoldbachgtde  34658  tgoldbachgtda  34659  tgoldbachgt  34661  prmdvdsfmtnof1lem1  47589  prmdvdsfmtnof  47591