Theorem isprm 10491

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 ∣ 𝑛 ∥ 𝑃} ≈ 2𝑜))

Distinct variable group:   𝑃,𝑛

Proof of Theorem isprm

Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 3789	. . . 4 ⊢ (𝑝 = 𝑃 → (𝑛 ∥ 𝑝 ↔ 𝑛 ∥ 𝑃))
2	1	rabbidv 2593	. . 3 ⊢ (𝑝 = 𝑃 → {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} = {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃})
3	2	breq1d 3795	. 2 ⊢ (𝑝 = 𝑃 → ({𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} ≈ 2𝑜 ↔ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} ≈ 2𝑜))
4		df-prm 10490	. 2 ⊢ ℙ = {𝑝 ∈ ℕ ∣ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} ≈ 2𝑜}
5	3, 4	elrab2 2751	1 ⊢ (𝑃 ∈ ℙ ↔ (𝑃 ∈ ℕ ∧ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} ≈ 2𝑜))

Syntax hints:   ∧ wa 102   ↔ wb 103   = wceq 1284   ∈ wcel 1433  {crab 2352   class class class wbr 3785  2_𝑜c2o 6018   ≈ cen 6242  ℕcn 8039   ∥ cdvds 10195  ℙcprime 10489

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rab 2357  df-v 2603  df-un 2977  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-prm 10490

This theorem is referenced by:  prmnn  10492  1nprm  10496  isprm2  10499