ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  prmssnn GIF version

Theorem prmssnn 11176
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
StepHypRef Expression
1 prmnn 11174 . 2 (𝑥 ∈ ℙ → 𝑥 ∈ ℕ)
21ssriv 3027 1 ℙ ⊆ ℕ
Colors of variables: wff set class
Syntax hints:  wss 2997  cn 8394  cprime 11171
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rab 2368  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-prm 11172
This theorem is referenced by:  prmex  11177
  Copyright terms: Public domain W3C validator