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Theorem isprm 16014
 Description: The predicate "is a prime number". A prime number is a positive integer with exactly two positive divisors. (Contributed by Paul Chapman, 22-Jun-2011.)
Assertion
Ref Expression
isprm (𝑃 ∈ ℙ ↔ (𝑃 ∈ ℕ ∧ {𝑛 ∈ ℕ ∣ 𝑛𝑃} ≈ 2o))
Distinct variable group:   𝑃,𝑛

Proof of Theorem isprm
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 breq2 5035 . . . 4 (𝑝 = 𝑃 → (𝑛𝑝𝑛𝑃))
21rabbidv 3427 . . 3 (𝑝 = 𝑃 → {𝑛 ∈ ℕ ∣ 𝑛𝑝} = {𝑛 ∈ ℕ ∣ 𝑛𝑃})
32breq1d 5041 . 2 (𝑝 = 𝑃 → ({𝑛 ∈ ℕ ∣ 𝑛𝑝} ≈ 2o ↔ {𝑛 ∈ ℕ ∣ 𝑛𝑃} ≈ 2o))
4 df-prm 16013 . 2 ℙ = {𝑝 ∈ ℕ ∣ {𝑛 ∈ ℕ ∣ 𝑛𝑝} ≈ 2o}
53, 4elrab2 3631 1 (𝑃 ∈ ℙ ↔ (𝑃 ∈ ℕ ∧ {𝑛 ∈ ℕ ∣ 𝑛𝑃} ≈ 2o))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {crab 3110   class class class wbr 5031  2oc2o 8086   ≈ cen 8496  ℕcn 11632   ∥ cdvds 15606  ℙcprime 16012 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-rab 3115  df-v 3443  df-un 3886  df-sn 4526  df-pr 4528  df-op 4532  df-br 5032  df-prm 16013 This theorem is referenced by:  prmnn  16015  1nprm  16020  isprm2  16023
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