Theorem prmssnn 16603

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 16601	. 2 ⊢ (𝑥 ∈ ℙ → 𝑥 ∈ ℕ)
2	1	ssriv 3937	1 ⊢ ℙ ⊆ ℕ

Syntax hints:   ⊆ wss 3901  ℕcn 12145  ℙcprime 16598

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-prm 16599

This theorem is referenced by:  prmex  16604  prminf  16843  prmgaplem3  16981  prmgaplem4  16982  prmdvdsfi  27073  mumul  27147  sqff1o  27148  dirith2  27495  hgt750lema  34814  tgoldbachgtde  34817  tgoldbachgtda  34818  tgoldbachgt  34820  prmdvdsfmtnof1lem1  47826  prmdvdsfmtnof  47828