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Theorem isprm 15667
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 4813 . . . 4 (𝑝 = 𝑃 → (𝑛𝑝𝑛𝑃))
21rabbidv 3338 . . 3 (𝑝 = 𝑃 → {𝑛 ∈ ℕ ∣ 𝑛𝑝} = {𝑛 ∈ ℕ ∣ 𝑛𝑃})
32breq1d 4819 . 2 (𝑝 = 𝑃 → ({𝑛 ∈ ℕ ∣ 𝑛𝑝} ≈ 2𝑜 ↔ {𝑛 ∈ ℕ ∣ 𝑛𝑃} ≈ 2𝑜))
4 df-prm 15666 . 2 ℙ = {𝑝 ∈ ℕ ∣ {𝑛 ∈ ℕ ∣ 𝑛𝑝} ≈ 2𝑜}
53, 4elrab2 3523 1 (𝑃 ∈ ℙ ↔ (𝑃 ∈ ℕ ∧ {𝑛 ∈ ℕ ∣ 𝑛𝑃} ≈ 2𝑜))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384   = wceq 1652  wcel 2155  {crab 3059   class class class wbr 4809  2𝑜c2o 7758  cen 8157  cn 11274  cdvds 15265  cprime 15665
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-br 4810  df-prm 15666
This theorem is referenced by:  prmnn  15668  1nprm  15672  isprm2  15675
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