Theorem isprm 12831

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 4118	. . . 4 ⊢ (𝑝 = 𝑃 → (𝑛 ∥ 𝑝 ↔ 𝑛 ∥ 𝑃))
2	1	rabbidv 2804	. . 3 ⊢ (𝑝 = 𝑃 → {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} = {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃})
3	2	breq1d 4124	. 2 ⊢ (𝑝 = 𝑃 → ({𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} ≈ 2o ↔ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} ≈ 2o))
4		df-prm 12830	. 2 ⊢ ℙ = {𝑝 ∈ ℕ ∣ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} ≈ 2o}
5	3, 4	elrab2 2979	1 ⊢ (𝑃 ∈ ℙ ↔ (𝑃 ∈ ℕ ∧ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} ≈ 2o))

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2205  {crab 2526   class class class wbr 4114  2_oc2o 6654   ≈ cen 6986  ℕcn 9254   ∥ cdvds 12498  ℙcprime 12829

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rab 2531  df-v 2817  df-un 3218  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-prm 12830

This theorem is referenced by:  prmnn  12832  1nprm  12836  isprm2  12839