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Theorem isprm 16614
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 5152 . . . 4 (𝑝 = 𝑃 → (𝑛𝑝𝑛𝑃))
21rabbidv 3440 . . 3 (𝑝 = 𝑃 → {𝑛 ∈ ℕ ∣ 𝑛𝑝} = {𝑛 ∈ ℕ ∣ 𝑛𝑃})
32breq1d 5158 . 2 (𝑝 = 𝑃 → ({𝑛 ∈ ℕ ∣ 𝑛𝑝} ≈ 2o ↔ {𝑛 ∈ ℕ ∣ 𝑛𝑃} ≈ 2o))
4 df-prm 16613 . 2 ℙ = {𝑝 ∈ ℕ ∣ {𝑛 ∈ ℕ ∣ 𝑛𝑝} ≈ 2o}
53, 4elrab2 3686 1 (𝑃 ∈ ℙ ↔ (𝑃 ∈ ℕ ∧ {𝑛 ∈ ℕ ∣ 𝑛𝑃} ≈ 2o))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1541  wcel 2106  {crab 3432   class class class wbr 5148  2oc2o 8462  cen 8938  cn 12216  cdvds 16201  cprime 16612
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-prm 16613
This theorem is referenced by:  prmnn  16615  1nprm  16620  isprm2  16623
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