Theorem prmssnn 16710

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 16708	. 2 ⊢ (𝑥 ∈ ℙ → 𝑥 ∈ ℕ)
2	1	ssriv 3999	1 ⊢ ℙ ⊆ ℕ

Syntax hints:   ⊆ wss 3963  ℕcn 12264  ℙcprime 16705

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-prm 16706

This theorem is referenced by:  prmex  16711  prminf  16949  prmgaplem3  17087  prmgaplem4  17088  prmdvdsfi  27165  mumul  27239  sqff1o  27240  dirith2  27587  hgt750lema  34651  tgoldbachgtde  34654  tgoldbachgtda  34655  tgoldbachgt  34657  prmdvdsfmtnof1lem1  47509  prmdvdsfmtnof  47511