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Theorem isprm 11958
 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 3965 . . . 4 (𝑝 = 𝑃 → (𝑛𝑝𝑛𝑃))
21rabbidv 2698 . . 3 (𝑝 = 𝑃 → {𝑛 ∈ ℕ ∣ 𝑛𝑝} = {𝑛 ∈ ℕ ∣ 𝑛𝑃})
32breq1d 3971 . 2 (𝑝 = 𝑃 → ({𝑛 ∈ ℕ ∣ 𝑛𝑝} ≈ 2o ↔ {𝑛 ∈ ℕ ∣ 𝑛𝑃} ≈ 2o))
4 df-prm 11957 . 2 ℙ = {𝑝 ∈ ℕ ∣ {𝑛 ∈ ℕ ∣ 𝑛𝑝} ≈ 2o}
53, 4elrab2 2867 1 (𝑃 ∈ ℙ ↔ (𝑃 ∈ ℕ ∧ {𝑛 ∈ ℕ ∣ 𝑛𝑃} ≈ 2o))
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   ↔ wb 104   = wceq 1332   ∈ wcel 2125  {crab 2436   class class class wbr 3961  2oc2o 6347   ≈ cen 6672  ℕcn 8812   ∥ cdvds 11660  ℙcprime 11956 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-rab 2441  df-v 2711  df-un 3102  df-sn 3562  df-pr 3563  df-op 3565  df-br 3962  df-prm 11957 This theorem is referenced by:  prmnn  11959  1nprm  11963  isprm2  11966
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