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Theorem isprm 16644
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 3437 . . 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 16643 . 2 ℙ = {𝑝 ∈ ℕ ∣ {𝑛 ∈ ℕ ∣ 𝑛𝑝} ≈ 2o}
53, 4elrab2 3685 1 (𝑃 ∈ ℙ ↔ (𝑃 ∈ ℕ ∧ {𝑛 ∈ ℕ ∣ 𝑛𝑃} ≈ 2o))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1534  wcel 2099  {crab 3429   class class class wbr 5148  2oc2o 8481  cen 8961  cn 12243  cdvds 16231  cprime 16642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-prm 16643
This theorem is referenced by:  prmnn  16645  1nprm  16650  isprm2  16653
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