Theorem prmssnn 16643

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 16641	. 2 ⊢ (𝑥 ∈ ℙ → 𝑥 ∈ ℕ)
2	1	ssriv 3926	1 ⊢ ℙ ⊆ ℕ

Syntax hints:   ⊆ wss 3890  ℕcn 12172  ℙcprime 16638

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-prm 16639

This theorem is referenced by:  prmex  16644  prminf  16884  prmgaplem3  17022  prmgaplem4  17023  prmdvdsfi  27095  mumul  27169  sqff1o  27170  dirith2  27516  hgt750lema  34848  tgoldbachgtde  34851  tgoldbachgtda  34852  tgoldbachgt  34854  prmdvdsfmtnof1lem1  48069  prmdvdsfmtnof  48071