Theorem prmssnn 16693

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 16691	. 2 ⊢ (𝑥 ∈ ℙ → 𝑥 ∈ ℕ)
2	1	ssriv 3940	1 ⊢ ℙ ⊆ ℕ

Syntax hints:   ⊆ wss 3904  ℕcn 12207  ℙcprime 16688

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-prm 16689

This theorem is referenced by:  prmex  16694  prminf  16934  prmgaplem3  17072  prmgaplem4  17073  prmdvdsfi  27148  mumul  27222  sqff1o  27223  dirith2  27569  hgt750lema  34915  tgoldbachgtde  34918  tgoldbachgtda  34919  tgoldbachgt  34921  prmdvdsfmtnof1lem1  48157  prmdvdsfmtnof  48159